Question : If $\cos^{2}\alpha-\sin^{2}\alpha=\tan^{2}\beta$, then the value of $\cos^{2}\beta-\sin^{2}\beta$ is:
Option 1: $\cot^{2}\alpha$
Option 2: $\cot^{2}\beta$
Option 3: $\tan^{2}\alpha$
Option 4: $\tan^{2}\beta$
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Correct Answer: $\tan^{2}\alpha$
Solution : Given: $\cos^{2}\alpha–\sin^{2}\alpha=\tan^{2}\beta$ We know that: $1+\tan^{2}\alpha=\sec^{2}\alpha$ So, $\cos^{2}\alpha-\sin^{2}\alpha=\sec^{2}\beta-1$ ⇒ $\cos^{2}\alpha-\sin^{2}\alpha+1=\sec^{2}\beta$ ⇒ $\cos^{2}\beta=\frac{1}{\cos^{2}\alpha–\sin^{2}\alpha+(\sin^{2}\alpha+\cos^{2}\alpha)}$ ⇒ $\cos^{2}\beta=\frac{1}{2\cos^{2}\alpha}$ Also, $\sin^{2}\beta=1-\cos^{2}\beta=1-\frac{1}{2\cos^{2}\alpha}$ So, $\cos^{2}\beta-\sin^{2}\beta$ $=\frac{1}{2\cos^{2}\alpha}-(1-\frac{1}{2\cos^{2}\alpha})$ $= \frac{1}{\cos^{2}\alpha}-1$ $= \sec^{2}\alpha-1$ $= \tan^{2}\alpha$ Hence, the correct answer is $\tan^{2}\alpha$.
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