Question : If $\cos^{4}\theta-\sin^{4}\theta=\frac{2}{3}$, then the value of $1-2\sin^{2}\theta$ is:
Option 1: $\frac{2}{3}$
Option 2: $\frac{3}{2}$
Option 3: $1$
Option 4: $0$
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Correct Answer: $\frac{2}{3}$
Solution :
$\cos^{4}\theta-\sin^{4}\theta=\frac{2}{3}$
⇒ $(\cos^{2}\theta-\sin^{2}\theta)(\cos^{2}\theta+\sin^{2}\theta) = \frac{2}{3}$
We know that, $\cos^{2}\theta+\sin^{2}\theta= 1$
⇒ $(\cos^{2}\theta-\sin^{2}\theta)=\frac{2}{3}$
Putting the value of $\cos^{2}\theta = 1-\sin^{2}\theta$
⇒ $(1-\sin^{2}\theta-\sin^{2}\theta)=\frac{2}{3}$
$\therefore(1-2\sin^{2}\theta)=\frac{2}{3}$
Hence, the correct answer is $\frac{2}{3}$.
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