Trigonometric Identities – Multiple Choice Questions with Solutions

This page contains multiple-choice questions on trigonometric identities. Each question is followed by a detailed explanation to help you understand why a statement is or is not an identity.

Questions

Question 1

Which of the following is not an identity?

a) \( \sin^2 a + \cos^2 a = 1 \)
b) \( \sin a = \tan a \cdot \cos a \)
c) \( 1 + \cot^2 a = \csc^2 a \)
d) \( 1 - \sec^2 a = \tan^2 a \)

Question 2

Which of the following is an identity?

a) \( \sin a \cos a = \frac{1}{2}\sin(2a) \)
b) \( \sin a + \cos a = 1 \)
c) \( \sin(-a) = \sin a \)
d) \( \tan a = \frac{\cos a}{\sin a} \)

Question 3

Simplify:

a) \( \sin t \)
b) \( \csc t \)
c) \( \sec t \)
d) \( \cos t \)

Question 4

Which of the following is not an identity?

a) \( \tan(2t) = 2\tan t \)
b) \( \sin^2 t = 1 - \cos^2 t \)
c) \( \sin(-t) = -\sin t \)
d) \( \sec(-t) = \sec t \)

Question 5

Which of the following is an identity?

a) \( \sin^2 u = 1 + \cos^2 u \)
b) \( \cot u = \sin u \cos u \)
c) \( \sin^2 u = 1 - \frac{1}{\sec^2 u} \)
d) \( \cos(-u) = -\cos u \)

Question 6

Simplify:

a) \( -\sin x \)
b) \( \sin x \)
c) \( \sec x \)
d) \( \cos x \)

Answers with Explanations

1. Answer: d)
   We know that: \[ 1 + \tan^2 a = \sec^2 a \] Rewriting, \[ 1 - \sec^2 a = -\tan^2 a \] So option (d) is false and therefore not an identity.
2. Answer: a)
   Using the double-angle identity: \[ \sin(2a) = 2\sin a \cos a \] Dividing both sides by 2 gives: \[ \sin a \cos a = \frac{1}{2}\sin(2a) \]
3. Answer: b)
   \[ \sin t + \frac{\cos^2 t}{\sin t} = \frac{\sin^2 t + \cos^2 t}{\sin t} = \frac{1}{\sin t} = \csc t \]
4. Answer: a)
   The correct identity for tangent is: \[ \tan(2t) = \frac{2\tan t}{1 - \tan^2 t} \] So \( \tan(2t) = 2\tan t \) is false.
5. Answer: c)
   Since \( \sec^2 u = \frac{1}{\cos^2 u} \), \[ 1 - \frac{1}{\sec^2 u} = 1 - \cos^2 u = \sin^2 u \]
6. Answer: a)
   Using angle-shift identities: \[ \sin(x - \pi) = -\sin x,\quad \sin(x + \pi) = -\sin x \] So: \[ \sin x - \sin x - \sin x = -\sin x \]

Further Practice

More Trigonometry Problems and Tutorials