Which of the following is/are identity/identities?1. 2. Select the correct answer using the code given below

Both 1 and 2. For statement 1: . Adding yields . Thus, statement 1 is an identity. For statement 2: . Expanding the right side factor: . Thus, statement 2 is also an identity.

Which of the following is/are identity/identities?<br>1. \frac{\sin^{3}\theta+\cos^{3}\theta}{\sin \theta+\cos \theta}+\sin \theta \cos \theta=1; 0 \lt \theta \lt \frac{\pi}{2}<br>2. 1-\sin^{6}\theta=\cos^{2}\theta(\cos^{4}\theta+3 \sin^{2}\theta)<br>Select the correct answer using the code given below

A. 1 only
B. 2 only
C. Both 1 and 2 ✓
D. Neither 1 nor 2

Correct Answer: C. Both 1 and 2

Explanation

For statement 1: \frac{\sin^3\theta+\cos^3\theta}{\sin\theta+\cos\theta} = \sin^2\theta - \sin\theta\cos\theta + \cos^2\theta = 1 - \sin\theta\cos\theta. Adding \sin\theta\cos\theta yields 1. Thus, statement 1 is an identity. For statement 2: 1-\sin^6\theta = (1-\sin^2\theta)(1+\sin^2\theta+\sin^4\theta) = \cos^2\theta(1+\sin^2\theta+\sin^4\theta). Expanding the right side factor: \cos^4\theta + 3\sin^2\theta = (1-\sin^2\theta)^2 + 3\sin^2\theta = 1 - 2\sin^2\theta + \sin^4\theta + 3\sin^2\theta = 1+\sin^2\theta+\sin^4\theta. Thus, statement 2 is also an identity.

Explanation

Related questions on Trigonometry