Question

If \[ (\sin\theta+\cos\theta)^4+(\sin\theta-\cos\theta)^4 = p - q(\sin^4\theta+\cos^4\theta), \] then \(p+q=\)

• A. 6
• B. 4
• C. 10
• D. 8 \bigskip

Hint:

Always reduce powers using standard symmetric identities first.

Answer 1

The Correct Option is C

Concept: Use identities: \[ (a+b)^4+(a-b)^4=2(a^4+b^4)+12a^2b^2 \]

Step 1: Apply identity.
Let \(a=\sin\theta, b=\cos\theta\) \[ =2(\sin^4\theta+\cos^4\theta)+12\sin^2\theta\cos^2\theta \]

Step 2: Convert using identity.
\[ \sin^2\theta\cos^2\theta=\frac{1-(\sin^4\theta+\cos^4\theta)}{2} \]

Step 3: Simplify.
\[ =2S+12\cdot\frac{1-S}{2} \] \[ =2S+6-6S \] \[ =6-4S \] Thus: \[ p=6,\quad q=4 \] \[ p+q=10 \] \[ \boxed{(C)} \]