Question : If $\cos ^4 \alpha-\sin ^4 \alpha=\frac{5}{6}$, then the value of $2 \cos ^2 \alpha-1$ is:
Option 1: $\frac{11}{6}$
Option 2: $\frac{5}{6}$
Option 3: $\frac{6}{11}$
Option 4: $\frac{6}{5}$
Correct Answer: $\frac{5}{6}$
Solution :
Given: $\cos ^4 \alpha-\sin ^4 \alpha=\frac{5}{6}$
⇒ $(\cos ^2 \alpha)^2-(\sin ^2 \alpha)^2=\frac{5}{6}$
⇒ $(\cos ^2 \alpha-\sin ^2 \alpha)(\cos ^2 \alpha+\sin ^2 \alpha)=\frac{5}{6}$
⇒ $(\cos ^2 \alpha-\sin ^2 \alpha)=\frac{5}{6}$ [As $\cos ^2 \alpha+\sin ^2 \alpha=1$]
⇒ $\cos^2\alpha - 1+\cos^2\alpha=\frac{5}{6}$ [As $\sin^2\alpha=1-\cos^2\alpha$]
$\therefore2\cos ^2 \alpha-1=\frac{5}{6}$
Hence, the correct answer is $\frac{5}{6}$.
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