ML Aggarwal Solution Class 10 Chapter 18 Trigonometric Identities MCQs

MCQs 

Choose the correct answer from the given four options (1 to 12) :

Question 1

cot2θ1sin2θ is equal to

(a) 1

(b) -1

(c) sin2θ

(d) sec2θ
Sol :
cot2θ1sin2θ
=cos2θsin2θ1sin2θ

cos2θ1sin2θ=sin2θsin2θ=1

Ans (b)


Question 2

(sec2θ1)(1cosec2θ) is equal to

(a) – 1

(b) 1

(c) 0

(d) 2

Sol :
(sec2θ1)(1cosec2θ)

=(1cos2θ1)(11sin2θ)

=1cos2θcos2θ×sin2θ1sin2θ

=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+(cosθsinθ)2 is equal to

(a) – 2

(b) 0

(c) 1

(d) 2

Sol :
(cosθ+sinθ)2+(cosθsinθ)2
=cos2θ+sin2θ+2sinθcosθ+cos2θ+sin2θ2sinθcosθ
=2(sin2θ+cos2θ)

=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)(1sinA)

=1+sinAcosA×1sinA

=(1+sinA)(1sinA)cosA

=1sin2AcosA=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

(a) 11k
(b) 1k
(c) 1+k
(d) 1k
Sol :

sec θ – tan θ = k

1cosθsinθcosθ=k

1sinθcosθ=k

Squaring both sides, we get

(1sinθcosθ)2=(k)2(1sinθ)2cos2θ=k2

=(1sinθ)21sin2θ=k2

(1sinθ)2(1+sinθ)(1sinθ)=k2

=1sinθ1+sinθ=k2

1+sinθ1sinθ=1k2

(1+sinθ)1sinθ=1k

Multiplying and dividing by (1+sinθ)

=[(1+sinθ)21sin2θ]=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°):

(a) cos2θsin2θ=1
(b) cosec2θsec2θ=1
(c) sec2θtan2θ=1
(c) cot2θtan2θ=1

Sol :

sec2θtan2θ=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 }

=sin2θ+cos2θ=1

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°

=sin225+cos225
(sin2θ+cos2θ=1)

=1
Ans (b)


Question 11

The value of 3tan2263cosec264 is

(a) 0

(b) 3

(c) – 3

(d) – 1

Sol :

3tan2263cosec264

=3tan2263cosec(9026)

=3tan2263sec226

=3(tan226sec226)

=3×(1)=3  {sec2θtan2θ=1}

Ans (c)


Question 12

The value of sin(90θ)sinθtanθ1 is

(a) cotθ

(b) sin2θ

(c) cos2θ

(d) cosec2θ

Sol :

sin(90θ)sinθtanθ1

=cosθsinθsinθcosθ1

=sinθcosθ×cosθsinθ1

=cos2θ1=(1cos2θ)

=sin2θ

Ans (b)

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