Trigonometry & Inverse Trigonometry Question - None

Question

If \sin^4 x + \cos^4 x = \sin x \cos x, then x is

Accepted Answer

Step 1: Use the identity \sin^4 x + \cos^4 x = 1 - 2\sin^2 x \cos^2 x

Step 2: So the equation becomes: 1 - 2\sin^2 x \cos^2 x = \sin x \cos x

Step 3: Let \sin x \cos x = t, then 1 - 2t^2 = t

Step 4: 2t^2 + t - 1 = 0 \Rightarrow (2t-1)(t+1) = 0

Step 5: t = \frac{1}{2} or t = -1

Step 6: For \sin x \cos x = \frac{1}{2}: \sin 2x = 1 \Rightarrow x = (4n+1)\frac{\pi}{4}

Step 7: For \sin x \cos x = -1: No real solution since \sin x \cos x \leq \frac{1}{2}

∴ Answer is C.

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