[CLASS 10] Mathematics MCQs on Trigonometric Identities

Mathematics Aptitude Test for Class 10 on “Trigonometric Identities – 2”.

1. If sec θ – tan θ = M then sec θ + tan θ = (frac {1}{M}).
a) False
b) True
Answer: b
Clarification: The appropriate trigonometric identity used here is sec² θ – tan² θ = 1.
(sec θ – tan θ) (sec θ + tan θ) = 1
M (sec θ + tan θ) = 1
(sec θ + tan θ) = (frac {1}{M})

2. Find the correct trigonometric identity.
a) cos² θ = 1 – sin² θ
b) cos² θ = 1 + sin² θ
c) tan² θ + sec² θ = 1
d) tan² θ = sec² θ + 1
Answer: a
Clarification: The appropriate trigonometric identity used here is sin² θ + cos² θ = 1.
cos² θ = 1 – sin² θ

3. Evaluate (sec θ – tan θ) (sec θ + tan θ).
a) 0
b) 1
c) 2
d) 3
Answer: b
Clarification: = (sec θ – tan θ) (sec θ + tan θ) = sec² θ – tan² θ
= 1
The identity used here is sec² θ – tan² θ = 1

4. Evaluate (sqrt {frac {1 – sin⁡ , ⁡A}{1 + sin , ⁡⁡A}}).
a) cos A + tan A
b) cos A – tan A
c) tan A – cot A
d) sec A + tan A
Answer: d
Clarification: (sqrt {frac {1 – sin⁡ , ⁡A}{1 + sin⁡ , ⁡A}} = sqrt {frac {1 – sin⁡ , ⁡A}{1 + sin , ⁡⁡A}} . sqrt {frac {1 – sin , ⁡A}{1 – sin , ⁡⁡A}})
= (frac {sqrt {(1 + sin , ⁡A)^2}}{sqrt {1 – sin^2} , ⁡A} )
= (frac {1 + sin⁡ , ⁡A}{sqrt {cos^2} , ⁡A})     (∵ sin² A + cos² A = 1)
= (frac {1 + sin⁡ , ⁡A}{cos⁡ , ⁡A})
= sec A + tan A

5. Evaluate (cosec² θ – cot² θ)² . (cosec θ + cot θ)².
a) 1
b) 0
c) (cosec² θ – cot² θ)²
d) (cosec θ + cot θ)²
Answer: d
Clarification: (cosec² θ – cot² θ)² . (cosec θ + cot θ)² = 1 (cosec θ + cot θ)²
= (cosec θ + cot θ)²

6. (1 – sin² A) (1 + tan² A) equals to _____
a) – Sec² θ Tan² θ
b) – Sec² θ Tan² θ
c) 1
d) 0
Answer: c
Clarification: (1 – sin² A) (1 + tan² A) = cos² A . sec² A
= cos² A . (frac {1}{cos^2 A})
= 1

7. Evaluate (cosec A – 1) (cosec A + 1) (sec² A – 1).
a) 0
b) 1
c) (frac {4}{3})
d) (frac {3}{4})
Answer: b
Clarification: (cosec A – 1) (cosec A + 1) (sec² A – 1) = (cosec² A – 1) (sec² A – 1)
= cot²A . tan² A
= (frac {1}{tan^2 A}) . tan² A
= 1

8. (sin A + cos A)² is equal to _____
a) 1 + 2sin A cos A
b) 1 – 2sin A cos A
c) 2sin A cos A – 1
d) 2sin A cos A + 1
Answer: a
Clarification: (sin A + cos A)² = sin² A + cos² A + 2sin A cos A
= 1 + 2sin A cos A

9. Evaluate tan² A + (1 + sec A) (sec A – 1).
a) 3 tan² A
b) 0
c) 2tan² A
d) 1
Answer: c
Clarification: tan² A + (1 + sec A) (sec A – 1) = tan² A + (sec A + 1) (sec A – 1)
= tan² A + (sec² A – 1)
= tan² A + tan² A
= 2tan² A

10. (1 + cosec⁡ θ) (1 – cosec⁡ θ) + cot²⁡ θ is _____
a) Cot ⁡θ
b) 0
c) 1
d) Tan ⁡θ
Answer: b
Clarification: (1 + cosec⁡ θ) (1 – cosec θ) + cot² ⁡θ = (1 – cosec² θ) + cot² θ
= -cot²⁡ θ + cot²⁡ θ     (∵ cosec² A – cot² A = 1)
= 0

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