ML Aggarwal Solution Class 10 Chapter 18 Trigonometric Identities MCQs
MCQs
Question 1
cot2θ−1sin2θ is equal to
(a) 1
(b) -1
cos2θ−1sin2θ=−sin2θsin2θ=−1
Ans (b)
Question 2
(sec2θ−1)(1−cosec2θ) is equal to
(a) – 1
(b) 1
(c) 0
(d) 2
=(1cos2θ−1)(1−1sin2θ)
=1−cos2θcos2θ×sin2θ−1sin2θ
=−sin2θcos2θsin2θcos2θ=−1
(∵sin2θ+cos2θ=1)
Ans (a)
Question 3
tan2θ1+tan2θ is equal to
(a) 2sin2θ
(b) 2cos2θ
(c) sin2θ
(d) cos2θ
Sol :
tan2θ1+tan2θ
tan2θ1+tan2θ
=sin2θcos2θ1+sin2θcos2θ
=sin2θcos2θcos2θ+sin2θcos2θ
=sin2θcos2θ×cos2θsin2θ+cos2θ
(∵sin2θ+cos2θ=1)
=sin2θ1=sin2θ
Ans (c)
Question 4
(cosθ+sinθ)2+(cosθ−sinθ)2 is equal to
(a) – 2
(b) 0
(c) 1
(d) 2
=2×1=2(d)
(∵sin2θ+cos2θ=1)
Question 5
(sec A + tan A) (1 – sin A) is equal to
(a) sec A
(b) sin A
(c) cosec A
(d) cos A
Sol :
(sec A + tan A) (1 – sin A)
=(1cosA+sinAcosA)(1−sinA)
=1+sinAcosA×1−sinA
=(1+sinA)(1−sinA)cosA
=1−sin2AcosA=cos2AcosA=cosA
Ans (d)
Question 6
1+tan2A1+cot2A is equal to
(a) sec2A
(b) -1
(c) cot2A
(d) tan2A
Sol :
1+tan2A1+cot2A
1+tan2A1+cot2A=1+sin2Acos2A1+cos2Asin2A
=cos2A+sin2Acos2Asin2A+cos2Asin2A=1cos2A1sin2A
=1cos2A×sin2A1
=sin2Acos2A=tan2A
Ans (d)
Question 7
If sec θ – tan θ = k, then the value of sec θ + tan θ is
sec θ – tan θ = k
1−sinθcosθ=k
Squaring both sides, we get
(1−sinθcosθ)2=(k)2⇒(1−sinθ)2cos2θ=k2
=(1−sinθ)21−sin2θ=k2
⇒(1−sinθ)2(1+sinθ)(1−sinθ)=k2
=1−sinθ1+sinθ=k2
⇒1+sinθ1−sinθ=1k2
⇒√(1+sinθ)1−sinθ=1k
Multiplying and dividing by (1+sinθ)
=√[(1+sinθ)21−sin2θ]=1k
⇒√[(1+sinθ)2cos2θ]=1k
1+sinθcosθ=1k=1cosθ+sinθcosθ=1k
⇒secθ+tanθ=1k
Ans (d)
Question 8
Which of the following is true for all values of θ (0° < θ < 90°):
Sol :
∴sec2θ−tan2θ=1 is true for all values of θ as it is an identity.
(0° < θ < 90°)
Ans (c)
Question 9
If θ is an acute angle of a right triangle, then the value of sin θ cos (90° – θ) + cos θ sin (90° – θ) is
(a) 0
(b) 2 sin θ cos θ
(c) 1
(c) 2sin2θ
Sol :
sin θ cos (90° – θ) + cos θ sin (90° – θ)
= sin θ sin θ + cos θ cos θ
{ ∵ sin(90° – θ) = cosθ, cos (90° – θ) = sin θ }
Ans (c)
Question 10
The value of cos 65° sin 25° + sin 65° cos 25° is
(a) 0
(b) 1
(b) 2
(d) 4
Sol :
cos 65° sin 25° + sin 65° cos 25°
= cos (90° – 25°) sin 25° + sin (90° – 25°) cos 25°
= sin 25° . sin 25° + cos 25° . cos 25°
Question 11
The value of 3tan226∘−3cosec264∘ is
(a) 0
(b) 3
(c) – 3
(d) – 1
Sol :
3tan226∘−3cosec264∘
=3tan226∘−3cosec(90∘−26∘)
=3tan226∘−3sec226∘
=3(tan226∘−sec226∘)
=3×(−1)=−3 {∵sec2θ−tan2θ=1}
Ans (c)
Question 12
The value of sin(90∘−θ)sinθtanθ−1 is
(a) −cotθ
(b) −sin2θ
(c) −cos2θ
(d) −cosec2θ
Sol :
sin(90∘−θ)sinθtanθ−1
=cosθsinθsinθcosθ−1
=sinθcosθ×cosθsinθ−1
=cos2θ−1=−(1−cos2θ)
=−sin2θ
Ans (b)
Comments
Post a Comment