ML Aggarwal Solution Class 9 Chapter 18 Trigonometric Ratios and Standard Angles Exercise 18.2

 Exercise 18.2

Question 1

Sol :

(i) Given : $\frac{\cos 18^{\circ}}{\sin 72^{\circ}}$

$\Rightarrow \frac{\cos 18^{\circ}}{\sin \left(90^{\circ}-18^{\circ}\right)}$ $\because \sin (90-\theta)=\cos \theta$

$\Rightarrow \frac{\cos 18^{\circ}}{\cos 18^{\circ}}$

=1

(ii) $\frac{\tan 41^{\circ}}{\cot 49^{\circ}}$

$\Rightarrow \frac{\tan 41^{\circ}}{\cot \left(90^{\circ}-41^{\circ}\right)}$ $\because \cot \left(90^{\circ}-\theta\right)=\tan \theta$

$\Rightarrow \frac{\tan 41^{\circ}}{\tan 41^{\circ}}$

=1

(iii) Given : $\frac{\operatorname{cosec} 17^{\circ} 30'}{\sec 72^{\circ} 30^{\prime}}$ $\sec (90-\theta)=\operatorname{cosec} \theta$

$\Rightarrow \frac{\operatorname{cosec} 17^{\circ}30'}{\sec \left(90^{\circ}-17^{\circ}30^{\prime}\right)}$

$\Rightarrow \quad \frac{\operatorname{cosec} 17^{\circ}30'}{\operatorname{cosec} 17^{\circ}30'}$

=1

Question 2

Sol :

(i) 

$\frac{\cot 40^{\circ}}{\tan 50^{\circ}}-\frac{1}{2}\left(\frac{\cos 35^{\circ}}{\sin 55^{\circ}}\right)$

$\Rightarrow \frac{\cot 40^{\circ}}{\tan \left(90-40^{\circ}\right)}-\frac{1}{2}\left[\frac{\cos 35^{\circ}}{\sin \left(90-35^{\circ}\right)}\right]$

$\Rightarrow \quad \frac{\cot 40^{\circ}}{\cot 40^{\circ}}-\frac{1}{2}\left(\frac{\cos 35^{\circ}}{\cos 35^{\circ}}\right)$

$\Rightarrow 1-\frac{1}{2}$

$\Rightarrow  \frac{2-1}{2}$

$\Rightarrow  \frac{1}{2}$

(ii) $\left(\frac{\sin 49^{\circ}}{\cos 41^{\circ}}\right)^{2}+\left(\frac{\cos 41^{\circ}}{\sin 49^{\circ}}\right)^{2}$

$\Rightarrow\left[\frac{\sin 49^{\circ}}{\cos \left(90-49^{\circ}\right)}\right]^{2}+\left[\frac{\cos 41^{\circ}}{\sin (90-41)}\right]^{2}$

$\Rightarrow\left(\frac{\sin 49^{\circ}}{\sin 49^{\circ}}\right)^{2}+\left(\frac{\cos 41^{\circ}}{\cos 41}\right)^{2}$

$\Rightarrow 1^{2}+1^{2}$

⇒1+1=2

(iii) $\frac{\sin 72^{\circ}}{\cos 18^{\circ}}-\frac{\sec 32^{\circ}}{\operatorname{cosec} 58^{\circ}}$

$\Rightarrow \frac{\sin 72^{\circ}}{\cos \left(90-12^{\circ}\right)}-\frac{\sec 32^{\circ}}{\operatorname{cosec}(90-32^{\circ})}$

$\Rightarrow \frac{\sin 72}{\sin 72^{\circ}}-\frac{\sec 32^{\circ}}{\operatorname{sec} 32^{\circ}}$

⇒1-1=0

(iv) $\frac{\cos 75^{\circ}}{\sin 15^{\circ}}+\frac{\sin 12^{\circ}}{\cos 78^{\circ}}-\frac{\cos 18^{\circ}}{\sin 12^{\circ}}$

$\Rightarrow \frac{\cos 75^{\circ}}{\sin \left(90-75^{\circ}\right)}+\frac{\sin 12^{\circ}}{\cos \left(90{-12^{\circ}}\right)}-\frac{\cos 18^{\circ}}{\sin \left(90-18^{\circ}\right)}$

$\Rightarrow \frac{\cos 75^{\circ}}{\cos 75^{\circ}}+\frac{\sin 12^{\circ}}{\operatorname{sin}12^{\circ}}-\frac{\cos 18^{\circ}}{\cos 18^{\circ}}$

⇒1+1-1

⇒1

(v) $\frac{\sin 25^{\circ}}{\sec 65^{\circ}}+\frac{\cos 25^{\circ}}{\operatorname{cosec} 65^{\circ}}$

$\Rightarrow \frac{\sin 25^{\circ}}{\sec (90-25)}+\frac{\cos 25^{\circ}}{\operatorname{cosec}\left(90-25^{\circ}\right)}$

$\Rightarrow \frac{\sin 25^{\circ}}{\operatorname{cosec} 25^{\circ}}+\frac{\cos 25^{\circ}}{\sec 25^{\circ}}$

$\Rightarrow \frac{\sin 25^{\circ}}{\frac{1}{\sin 25^{\circ}}}+\frac{\cos 25^{\circ}}{\frac{1}{\cos 25^{\circ}}}$

$\Rightarrow  \sin ^{2} 25^{\circ}+\cos ^{2} 25^{\circ}$ $\left(\because \cos ^{2} \theta+\operatorname{sin}^{2} \theta=1\right)$

⇒1

Question 3

(i) $\sin 62^{\circ}-\cos 28^{\circ}$

$\Rightarrow \sin 62^{\circ}-\cos \left(90-62^{\circ}\right)$

$\Rightarrow \sin 62^{\circ}-\sin 62^{\circ}$

⇒0

(ii) $\operatorname{cosec} 35^{\circ}-\sec 55^{\circ}$

$\Rightarrow  \operatorname{cosec} 35^{\circ}-\sec \left(90-35^{\circ}\right)$

$\Rightarrow  \operatorname{cosec} 35^{\circ}-\operatorname{cosec} 35^{\circ}$

⇒0

Question 4

Sol :

(i) $\cos ^{2} 26^{\circ}+\cos 64^{\circ} \sin 26^{\circ}+\frac{\tan 36^{\circ}}{\cot 54^{\circ}}$

$\Rightarrow \cos ^{2} 26^{\circ}+\cos \left(90-26^{\circ}\right) \sin 26^{\circ}+\frac{\tan 36^{\circ}}{\cot \left(90-36^{\circ}\right)}$

$\Rightarrow \cos ^{2} 26^{\circ}+\sin 26^{\circ} \sin 26^{\circ}+\frac{\tan 36^{\circ}}{\tan 36^{\circ}}$

$\Rightarrow \cos ^{2} 26^{\circ}+\sin ^{2} 26+1$

⇒1+1=2

(ii) $\frac{\sec 17^{\circ}}{\operatorname{cosec} 73^{\circ}}+\frac{\tan 68^{\circ}}{\cot 22^{\circ}}+\cos ^{2} 44^{\circ}+\cos ^{2} 46^{\circ}$

$\Rightarrow \frac{\sec 17^{\circ}}{\operatorname{cosec}(90-17)}+\frac{\tan 68^{\circ}}{\cot \left(90-68^{\circ}\right)}+\cos ^{2} 46^{\circ}+\cos ^{2} 44^{\circ}$

$\Rightarrow \frac{\sec 17^{\circ}}{\sec 17^{\circ}}+\frac{\tan 68^{\circ}}{\tan 68^{\circ}}+\cos ^{2}\left(90^{\circ}-44^{\circ}\right)+\cos ^{2} 44^{\circ}$

$\Rightarrow 1+1+\left(\sin ^{2} 44^{\circ}+\cos ^{2} 44\right)$

⇒2+1=3

Question 5

Sol :

(i) $\frac{\sin 65^{\circ}}{\cos 25^{\circ}}+\frac{\cos 32^{\circ}}{\sin 58^{\circ}}-\sin 28^{\circ} \sec 62^{\circ}+\operatorname{cosec}^{2} 30^{\circ}$

$\Rightarrow \frac{\sin 65^{\circ}}{\cos \left(90-65^{\circ}\right)}+\frac{\cos 32^{\circ}}{\sin \left(90-32^{\circ}\right)}-\sin 28 \sec \left(90-28^{\circ}\right)+(2)^{2}$

$\Rightarrow  \frac{\sin 65^{\circ}}{\sin 65^{\circ}}+\frac{\cos 32^{\circ}}{\cos 32^{\circ}}-\sin 28 \cdot \cos 28+4$

$\Rightarrow 1+1-\sin28 \frac{1}{\sin 28}+4$

⇒6-1=5

(ii) $\frac{\sec 29^{\circ}}{\operatorname{cosec} 61^{\circ}}+2 \cot 8^{\circ} \cdot \cot 17^{\circ} \cdot \cot 45^{\circ} \cdot \cot 73^{\circ} \cdot \cot 82^{\circ}-3\left(\sin ^{2} 38^{\circ}+\sin ^{2} 52^{\circ}\right)$

$\Rightarrow \frac{\sec 29^{\circ}}{\operatorname{cosec}\left(90-29^{\circ}\right)}+2 \cot \left(90-82^{\circ}\right) \cdot \cot \left(90-73^{\circ}\right) \cot 45^{\circ}\cot 73^{\circ} \cot 82^{\circ}-3\left(\sin ^{2} 38^{\circ}+\sin ^{2}\left(90-38^{\circ}\right)\right)$

$\Rightarrow \frac{\sec 29^{\circ}}{\operatorname{cosec} 29^{\circ}}+2 \cot 82^{\circ} \tan 82^{\circ} \cdot \cot 73^{\circ} \cdot \tan 73^{\circ} \cot 45^{\circ}-3\left(\sin ^{2} 38^{\circ}+\cos ^{2} 38^{\circ}\right)$

⇒1+2(1)-3(1)

⇒1+2-3=0

Question 6

Sol :

(i) $\tan 81^{\circ}+\cos 72^{\circ}$

⇒tan(90-9°)+cos(90-18°)

⇒cot 9°+sin 18°

(ii) $\Rightarrow \cot 49^{\circ}+\operatorname{cosec} 87^{\circ}$

$\Rightarrow \cot \left(90-41^{\circ}\right)+\operatorname{cosec}\left(90-5^{\circ}\right)$

$\Rightarrow \tan 41^{\circ}+\sec 3^{\circ}$

Question 7

Sol :

(i) $\sin ^{2} 28^{\circ}-\cos ^{2} 62^{\circ}=0$

$\Rightarrow  \sin ^{2} 28-\cos ^{2} 62^{\circ}$

$\Rightarrow \sin ^{2} 28^{\circ}-\cos ^{2}\left(90-28^{\circ}\right)$

$\Rightarrow \sin ^{2} 28^{\circ}-\sin ^{2} 28^{\circ}$

⇒0

∴LHS=RHS

(ii) LHS$\cos ^{2} 25^{\circ}+\cos ^{2} 65^{\circ}$

$\cos ^{2} 25^{\circ}+\cos ^{2}\left(90-25^{\circ}\right)$

$\cos ^{2} 25^{\circ}+\sin ^{2} 25$

⇒1

=RHS

∴LHS=RHS

(iii) $\Rightarrow \operatorname{cosec}^{2} 67^{\circ}-\tan ^{2} 23^{\circ}$

$\Rightarrow \operatorname{cosec}^{2}\left(90-23^{\circ}\right)-\tan ^{2} 23^{\circ}$

$\Rightarrow \sec ^{2} 23^{\circ}-\tan ^{2} 23^{\circ}$

⇒1

∴LHS=RHS

(iv) LHS $\sec ^{2} 22^{\circ}-\cot ^{2} 68^{\circ}$

$\Rightarrow \sec ^{2} 22^{\circ}-\cot ^{2}\left(90-22^{\circ}\right)$

$\Rightarrow \sec ^{2} 22^{\circ}-\tan ^{2} 22^{\circ}$

⇒1=RHS

Question 8

Sol :

(i) LHS$\Rightarrow \sin 63^{\circ} \cdot \cos 27^{\circ}+\cos 63^{\circ} \sin 27^{\circ}$

$\Rightarrow \sin 63^{\circ} \cos \left(90-63^{\circ}\right)+\cos 63^{\circ} \sin \left(90-63^{\circ}\right)$

$\Rightarrow \quad \sin 63^{\circ} \sin 63^{\circ}+\cos 63^{\circ} \cdot \cos 63^{\circ}$

$\Rightarrow \sin ^{2} 63^{\circ}+\cos ^{2} 63^{\circ}$

⇒1

∴LHS=RHS

(ii) LHS$\sec 31^{\circ} \sin 59^{\circ}+\cos 31^{\circ} \operatorname{cosec} 59^{\circ}$

$\Rightarrow \sec 31^{\circ} \sin \left(90-31^{\circ}\right)+\cos 31^{\circ} \operatorname{cosec}\left(90-31^{\circ}\right)$

$\Rightarrow \sec 31^{\circ} \cos 31^{\circ}+\cos 31^{\circ} \sec 31^{\circ}$

$\Rightarrow 2 \sec 31^{\circ} \cos 31^{\circ}$

$\Rightarrow 2 \frac{1}{\cos 31^{\circ}}.\cos 31^{\circ}$

⇒2

∴LHS=RHS

Question 9

Sol :

(i) LHS$\Rightarrow \sec 70^{\circ} \sin 20^{\circ}-\cos 20^{\circ} \cdot \operatorname{cossec} 70^{\circ}$

$\Rightarrow \sec 70^{\circ} \sin \left(90-70^{\circ}\right)-\cos 20^{\circ} \operatorname{cosec}\left(90-20^{\circ}\right)$

$\Rightarrow \sec 70^{\circ} \cdot \cos 20^{\circ}-\cos 28^{\circ} \sec 70^{\circ}$

⇒0

(ii) LHS$\Rightarrow \sin ^{2} 20^{\circ}+\sin ^{2} 70^{\circ}-\tan ^{2} 45$

$\Rightarrow  \sin ^{2} 20^{\circ}+\sin ^{2}\left(90-20^{\circ}\right)-\tan ^{2} 45^{\circ}$

$\Rightarrow \sin ^{2} 20+\cos ^{2} 20^{\circ}-(1)^{2}$

⇒1-1

⇒0

∴LHS=RHS

Question 10

Sol :

(i) LHS$\Rightarrow \frac{\cot 54^{\circ}}{\tan 36^{\circ}}+\frac{\tan 20^{\circ}}{\cot 70^{\circ}}-2$

$\Rightarrow \frac{\cot 54^{\circ}}{\tan (90-54)}+\frac{\tan 20^{\circ}}{\cot \left(90{-20}\right)}-2$

$\Rightarrow \frac{\cot 54^{\circ}}{\cot 54^{\circ}}+\frac{\tan 20^{\circ}}{\tan 20^{\circ}}-2$

⇒1+1-2

⇒0

∴LHS=RHS

(ii) LHS$\Rightarrow \frac{\cos 80^{\circ}}{\sin 10^{\circ}}+\cos 59^{\circ} \operatorname{cosec} 31^{\circ}$

$\Rightarrow \frac{\cos 80^{\circ}}{\sin \left(90-80^{\circ}\right)}+\cos 59^{\circ} \operatorname{cosec}\left(90-31\right)^{\circ}$

$\Rightarrow \frac{\cos 80^{\circ}}{\cos 80^{\circ}}+\cos 89^{\circ} \sec 59^{\circ}$

⇒1+1=2=RHS

Question 12

Sol :

(i) $2\left(\frac{\tan 35^{\circ}}{\cot 55^{\circ}}\right)^{2}+\left(\frac{\cot 55^{\circ}}{\tan 35^{\circ}}\right)-3\left(\frac{\sec 40^{\circ}}{\operatorname{cosec} 50^{\circ}}\right)$

$\Rightarrow$ 2. $\left[\frac{\tan 35^{\circ}}{\cot \left(90-35^{\circ}\right)}\right]^{2}+\frac{\cot 55^{\circ}}{\tan \left(90-55^{\circ}\right)}-3\left(\frac{\sec 40^{\circ}}{cosec(90-40^{\circ})}\right)$

$\Rightarrow 2\left(\frac{\tan 55^{\circ}}{\operatorname{tan} 35^{\circ}}\right)^{2}+\frac{\cot 55}{\cot 55}-3 \frac{\sec 40}{\sec 40}$

⇒2+1-3

⇒0

(ii) $\frac{\sin 35^{\circ} \cos 55^{\circ}+\cos 35^{\circ} \sin 55^{\circ}}{\operatorname{cosec}^{2} 10^{\circ}-\tan ^{2} 80}$

$\Rightarrow \frac{\sin 35^{\circ} \cdot \cos \left(90-35^{\circ}\right)+\cos 35^{\circ} \sin \left(90-35^{\circ}\right)}{\operatorname{cosec}^{2}\left(90-80^{\circ}\right)-\tan ^{2} 80^{\circ}}$

$\Rightarrow \frac{\sin 35^{\circ} \cdot \sin 35^{\circ}+\cos 35^{\circ} \cdot \cos 35^{\circ}}{\sec ^{2} 80^{\circ}+\tan ^{2} 80^{\circ}}$

$\frac{\sin ^{2} 35^{\circ}+\cos ^{2} 35^{\circ}}{\sec ^{2} 80-\tan ^{2} 80}$

⇒$\frac{1}{1}$=1

(iii) $\sin ^{2} 34^{\circ}+\sin ^{2} 56^{\circ}+2 \tan 18^{\circ} \cdot \tan 72^{\circ}-\cot ^{2} 30^{\circ}$

$\Rightarrow \sin ^{2} 34^{\circ}+\sin ^{2}\left(90{-34}\right)+2 \tan 18 \cdot \tan \left(90-18^{\circ}\right)-\cot ^{2} 30$

⇒1+2-3=0

Question 13

Sol :

(i) LHS$\Rightarrow \frac{\cos \theta}{\sin (90-\theta)}+\frac{\sin \theta}{\cos (90-\theta)}$

$\Rightarrow \frac{\cos \theta}{\cos \theta}+\frac{\sin \theta}{\sin \theta}$

⇒1+1=2

∴LHS=RHS

(ii) L H S$\Rightarrow \cos \theta \cdot \sin (90-\theta)+\sin \theta \cdot \cos (90-\theta)$

$\Rightarrow \cos \theta \cdot \cos \theta+\sin \theta \cdot \sin \theta$

$\Rightarrow \cos ^{2} \theta+\sin ^{2} \theta$

⇒1

∴LHS=RHS

(iii) LHS=$\frac{\tan \theta}{\tan (90-\theta)}+\frac{\sin (90-\theta)}{\cos \theta}$

$\Rightarrow \frac{\tan \theta}{\cot \theta}+\frac{\cos \theta}{\cos \theta}$

$\Rightarrow \frac{\tan \theta}{\left(\frac{1}{\tan \theta}\right)}+1$

⇒$\tan ^{2} \theta+1$ $\left(\because \sec ^{2} \theta-\tan ^{2} \theta=1\right)$

⇒$\sec ^{2} \theta$

∴LHS=RHS

Question 14

Sol :

(i) LHS⇒$\frac{\cos \left(90^{\circ}-A\right) \cdot \sin \left(90^{\circ}-A\right)}{\tan \left(90^{\circ}-A\right)}$

⇒$\frac{\sin A \cdot \cos A}{\cot A}$

$\Rightarrow \frac{\sin A \cdot \cos A}{\frac{\cos A}{\sin A}}$ $\left(\because \sin ^{2} A+\cos ^{2} A=1\right)$

$\Rightarrow \sin ^{2} A$

$\Rightarrow 1-\cos ^{2} A$

(ii) LHS$\Rightarrow \frac{\sin (90^{\circ}-A)}{\operatorname{cosec}\left(90^{\circ}-A\right)}+\frac{\cos (90^{\circ}-A)}{\sec \left(90^{\circ}-A\right)}$

$\Rightarrow \frac{\sin \cos A}{\sec A}+\frac{\sin A}{\operatorname{cosec} A}$

$\Rightarrow   \frac{\cos A}{\left(\frac{1}{\cos A}\right)}+\frac{\sin A}{\left(\frac{1}{\sin A}\right)}$

$\Rightarrow \cos A \cdot \cos A+\sin A \cdot \sin A$

$\Rightarrow \cos ^{2} A+\sin ^{2} A$

⇒1

∴LHS=RHS

Question 15

Sol :

(i) $\frac{\cos \theta}{\sin (90-\theta)}+\frac{\cos \left(90^{\circ}-\theta\right)}{\sec \left(90^{\circ}-\theta\right)}-3 \tan ^{2} 30^{\circ}$

$\Rightarrow \frac{\cos \theta}{\cos \theta}+\frac{\sin A}{\operatorname{cosec} \theta}-3\left(\frac{1}{\sqrt{3}}\right)^{2}$

⇒$1+\frac{\sin A}{\left(\frac{1}{\sin A}\right)}-3 \frac{1}{3}$

⇒$1+\sin ^{2} A-1$

⇒$\sin ^{2} A$

(ii) $\frac{\operatorname{cosec}\left(90^{\circ}-\theta\right) \cdot \sin \left(90^{\circ}-\theta\right) \cdot \cot \left(90^{\circ}-\theta\right)}{\cos \left(90^{\circ}-\theta\right) \cdot \sec \left(90^{\circ}-\theta\right) \cdot \tan \theta}+\frac{\cot \theta}{\tan (90-\theta)}$

$\Rightarrow \frac{\sec \theta \cdot \cos \theta \cdot \tan \theta}{\sin \theta \cdot \operatorname{cosec} \theta \cdot \tan \theta}+\frac{\cot \theta}{\cot \theta}$

$\frac{\frac{1}{\cos \theta} \cdot \cos \theta}{\sin \theta \frac{1}{\sin \theta}}+1$

⇒1+1

⇒2

Question 16

Sol :

(i) $\cos 63^{\circ} \cdot \sec \left(90^{\circ}-\theta\right)=1$

$\Rightarrow \cos 63^{\circ} \operatorname{cosec} \theta=1$

$\Rightarrow \cos 63^{\circ}=\frac{1}{\operatorname{cosec} \theta} .$

$\Rightarrow \cos 63^{\circ}=\sin \theta$

$\Rightarrow \cos 63^{\circ}=\cos (90-\theta)$

∴90-θ=63°

θ=90°-63°=27°

(ii) $\operatorname{tan} 35^{\circ} \cdot \cot (90-\theta)=1$

$\cot \left(90^{\circ}-\theta\right)=\frac{1}{\operatorname{tan} 35^{\circ}}$

$\cot \left(90^{\circ}-\theta\right)=\cot 35^{\circ}$

$90^{\circ}-\theta=35^{\circ}$

$\theta=90^{\circ}-35^{\circ}$

$\theta=55^{\circ}$

(ii) LHS

$\Rightarrow 4\left(\sin ^{4} 30^{\circ}+\cos ^{4} 60^{\circ}\right)-3\left(\cos ^{2} 45^{\circ}-\sin ^{2} 90^{\circ}\right)$

$\Rightarrow 4 \cdot\left[\left(\frac{1}{2}\right)^{4}+\left(\frac{1}{2}\right)^{4}\right]-3\left(\left(\frac{1}{\sqrt{2}}\right)^{2}-1^{2}\right)$

$\Rightarrow 4\left(\frac{1}{16}+\frac{1}{16}\right)-3\left(\frac{1}{2}-1\right)$

$\Rightarrow 4\left(\frac{1+1}{16}\right)-3\left(-\frac{1}{2}\right)$

$\Rightarrow 4 \frac{2}{16}+\frac{3}{2}$

$\Rightarrow \frac{1}{2}+\frac{3}{2}$

$\Rightarrow \frac{1}{2}+\frac{3}{2}$

$\Rightarrow \frac{1+3}{2}$

$\Rightarrow \frac{-4}{2}$

⇒-2=RHS

(iii) LHS⇒$\cos 60^{\circ}=\frac{1}{2}$

RHS$\Rightarrow \cos ^{2} 30^{\circ}-\sin ^{2} 30^{\circ}=\left(\frac{\sqrt{3}}{2}\right)^{2}-\left(\frac{1}{2}\right)^{2}$

$=\frac{3}{4}-\frac{1}{4}$

$=\frac{3-1}{4}$

$=\frac{2}{4}$

$=\frac{1}{2}$

∴LHS=RHS