Trigonometric Ratios and Identities
Algebraic Simplification
JEE Main 2019
Grade 11
Question:
For $\theta \in (\frac{\pi}{4}, \frac{\pi}{2})$, the expression $3(\cos \theta - \sin \theta)^4 + 6(\sin \theta + \cos \theta)^2 + 4 \sin^6 \theta$ equals:
(1) $13 - 4 \cos^4 \theta$
(2) $13 - 4 \cos^6 \theta$
(3) $13 + 4 \cos^6 \theta$
(4) $13 + 4 \cos^4 \theta$
Step-by-Step Solution
Key Concept: Let $a = \cos \theta - \sin \theta$, $b = \cos \theta + \sin \theta$. Note $a^2 + b^2 = 2$. Expand $3a^4$ using $a^2 = 1 - \sin 2\theta$, simplify, then factor $4 \sin^6 \theta$ as $4(1 - \cos^2 \theta)^3$.
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Correct Answer: (2)