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# If ${\tan ^2}\theta = 2{\tan ^2}\phi + 1$ . Then find the value of $\cos 2\theta + {\sin ^2}\phi .$ A) -1B) 0C) 1D) None of these

Last updated date: 20th Jun 2024
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Hint:To solve the problem we have to find the value of $\cos 2\theta$ in terms of $\phi$ . Then we have to calculate the value of $\cos 2\theta + {\sin ^2}\phi .$

From the given equation we have ${\tan ^2}\theta = 2{\tan ^2}\phi + 1$.
We know that $\cos 2\theta = \dfrac{{1 - {{\tan }^2}\theta }}{{1 + {{\tan }^2}\theta }}$ .
Let us substitute the value of ${\tan ^2}\theta$ in the above formula, we get,
$\cos 2\theta = \dfrac{{1 - {{\tan }^2}\theta }}{{1 + {{\tan }^2}\theta }} = \dfrac{{1 - (2{{\tan }^2}\phi + 1)}}{{1 + (2{{\tan }^2}\phi + 1)}}.$
Now we can simplify the above equation to get the following equation, $\cos 2\theta = \dfrac{{ - 2{{\tan }^2}\phi }}{{2 + 2{{\tan }^2}\phi }}.$
Taking 2 common from denominator, $\cos 2\theta = \dfrac{{ - 2{{\tan }^2}\phi }}{{2(1 + {{\tan }^2}\phi )}}.$
Cancelling the common factor 2 from both numerator and denominator, we get,
$\cos 2\theta = \dfrac{{ - {{\tan }^2}\phi }}{{1 + {{\tan }^2}\phi }}.$
We know that ${\tan ^2}\phi = \dfrac{{{{\sin }^2}\phi }}{{{{\cos }^2}\phi }}$ therefore, we get,
$\cos 2\theta = \dfrac{{ - {{\tan }^2}\phi }}{{1 + {{\tan }^2}\phi }} = - \dfrac{{\dfrac{{{{\sin }^2}\phi }}{{{{\cos }^2}\phi }}}}{{1 + \dfrac{{{{\sin }^2}\phi }}{{{{\cos }^2}\phi }}}}$
Further simplifying the above equation we get, $\cos 2\theta = - \dfrac{{\dfrac{{{{\sin }^2}\phi }}{{{{\cos }^2}\phi }}}}{{\dfrac{{{{\cos }^2}\phi + {{\sin }^2}\phi }}{{{{\cos }^2}\phi }}}}$
We now that ${\cos ^2}\phi + {\sin ^2}\phi = 1$substitute the identity in the above equation, we get,
$\cos 2\theta = - \dfrac{{\dfrac{{{{\sin }^2}\phi }}{{{{\cos }^2}\phi }}}}{{\dfrac{1}{{{{\cos }^2}\phi }}}}$
Further simplifying the values we get,
$\cos 2\theta = - \dfrac{{{{\sin }^2}\phi }}{{{{\cos }^2}\phi }} \times \dfrac{{{{\cos }^2}\phi }}{1} = - {\sin ^2}\phi .$
Now we calculate the value of $\cos 2\theta + {\sin ^2}\phi .$
Putting the value of$\cos 2\theta = - {\sin ^2}\phi$, we obtain,
$\cos 2\theta + {\sin ^2}\phi = - {\sin ^2}\phi + {\sin ^2}\phi = 0.$
$\cos 2\theta + {\sin ^2}\phi$= 0
Hence we have come to the conclusion that the correct answer is option (B).

Note: $\sin \theta$ , $\cos \theta$ , $\tan \theta$ etc. are called circular angles. These are the ratios of three sides of a right angle triangle taking two sides at a time. The ratio of perpendicular and hypotenuse is $\sin \theta$, the ratio of base and hypotenuse is $\cos \theta$ and the ratio of perpendicular and base is $\tan \theta$. The reciprocal of $\sin \theta$ is $\co sec\theta$ , the reciprocal of $\cos \theta$ is $\sec \theta$ and the reciprocal of $\tan \theta$ is $\cot \theta .$ Also $\tan \theta$ is the ratio of $\sin \theta$ and $\cos \theta$, $\cot \theta$ is the ratio of $\cos \theta$ and $\sin \theta$ . If an angle be $\theta$ , then $2\theta$ is the multiple angle. So $\sin 2\theta ,\cos 2\theta$ are the sine and cosine formulae for multiple angles.