Question : If $\cos\theta - \sin\theta =0$, then $(\sin^8\theta+\cos^8\theta)$ is:
Option 1: $\frac{1}{8}$
Option 2: $ \frac{1}{4}$
Option 3: $\frac{1}{6}$
Option 4: $\frac{1}{2}$
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Correct Answer: $\frac{1}{8}$
Solution :
Given: $\cos\theta - \sin\theta =0$
⇒ $\sin\theta = \cos\theta$
$\sin\theta$'s value is only equal with $\cos\theta$ when $\theta = 45°$
⇒ $\sin\theta = \cos\theta = \frac{1}{\sqrt2}$
$(\sin^8\theta+\cos^8\theta)$
= $(\sin^845°+\cos^845°)$
= $(\frac{1}{\sqrt2})^8+ (\frac{1}{\sqrt2})^8$
= $\frac{1}{16}+\frac{1}{16}$
= $\frac{2}{16}$
= $\frac{1}{8}$
Hence, the correct answer is $\frac{1}{8}$.
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