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If $\tan ^{2} \theta=2 \tan ^{2} \phi+1$, then which one of the following is correct?
Options:
Solution:
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Verified Answer
The correct answer is:
$\cos (2 \theta)=[\cos (2 \phi)-1] / 2$
Work with option,
$\cos (2 \phi)-1=\frac{1-\tan ^{2} \phi}{1+\tan ^{2} \phi}-1$
$=-\frac{2 \tan ^{2} \phi}{1+\tan ^{2} \phi}=\frac{-\left(\tan ^{2} \theta-1\right)}{1+\frac{\tan ^{2} \theta-1}{2}}$
$=\frac{1-\tan ^{2} \theta}{1+\tan ^{2} \theta} \times 2=\cos (2 \theta) 2$
Thus, $\cos 2 \theta=\frac{\cos (2 \phi)-1}{2}$
$\cos (2 \phi)-1=\frac{1-\tan ^{2} \phi}{1+\tan ^{2} \phi}-1$
$=-\frac{2 \tan ^{2} \phi}{1+\tan ^{2} \phi}=\frac{-\left(\tan ^{2} \theta-1\right)}{1+\frac{\tan ^{2} \theta-1}{2}}$
$=\frac{1-\tan ^{2} \theta}{1+\tan ^{2} \theta} \times 2=\cos (2 \theta) 2$
Thus, $\cos 2 \theta=\frac{\cos (2 \phi)-1}{2}$
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