Question : If $\cos ^2 \theta-\sin ^2 \theta=\tan ^2 \phi$, then which of the following is true?
Option 1: $\cos \theta \cos \phi=1$
Option 2: $\cos ^2 \phi-\sin ^2 \phi=\tan ^2 \theta$
Option 3: $\cos ^2 \phi-\sin ^2 \phi=\cot ^2 \theta$
Option 4: $\cos \theta \cos \phi=\sqrt{2}$
Correct Answer: $\cos ^2 \phi-\sin ^2 \phi=\tan ^2 \theta$
Solution :
Given,
$\cos^2\theta – \sin^2 \theta = \tan^2 \phi$
⇒ $\frac{\cos^2 \theta – \sin^2\theta}{1} = \frac{\sin^2\phi}{\cos^2 \phi}$
⇒ $\frac{\cos^2 \theta – \sin^2 \theta}{\cos^2\theta + \sin^2 \theta} = \frac{\sin^2\phi}{\cos^2 \phi}$
By Componendo and Dividendo,
⇒ $\frac{\cos^2 \theta}{–\sin^2 \theta} = \frac{\sin^2 \phi + \cos^2 \phi}{\sin^2 \phi - \cos^2 \phi}$
⇒ $\frac{–\sin^2\theta}{\cos^2 \theta} = \frac{\sin^2 \phi – \cos^2 \phi}{\sin^2 \phi+ \cos^2 \phi}$
⇒ $\frac{–\sin^2\theta}{\cos^2 \theta}= \frac{\sin^2\phi – \cos^2 \phi}{1}$ [As $\sin^2\phi + \cos^2\phi=1$]
⇒ $\tan^2 \theta = \cos^2\phi-\sin^2\phi $
Hence, the correct answer is $\tan^2 \theta = \cos^2 \phi-\sin^2 \phi$.
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