Question

If \( \sin x + \sin^2 x = 1 \), \( x \in \left(0, \frac{\pi}{2} \right) \), then the expression \[ (\cos^2 x + \tan^2 x) + 3(\cos^4 x + \tan^4 x + \cos^4 x + \tan^4 x) + (\cos^6 x + \tan^6 x) \] is equal to:

• A. 4
• B. 3
• C. 2
• D. 1

Hint:

For trigonometric identities, converting all terms into sine and cosine often simplifies the calculation effectively.

Answer 1

The Correct Option is C

Step 1: Understanding the given identity. Given \( \sin x + \sin^2 x = 1 \), we have: \[ \sin x = \cos^2 x \quad \Rightarrow \quad \tan x = \cos x \]

Step 2: Expanding the given expression. The given expression becomes: \[ 2\cos^2 x + [\cos^6 x + \cos^3 x] + 2\cos^6 x \] \[ = 2[\sin^2 x + \sin^3 x + \sin^4 x] \] \[ = 2\sin^2 x [( \sin x + 1 )^2] \] \[ = 2[\sin^2 x + \sin^3 x] = 2 \]