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Q.

If $\tan A + \sin A = m a n d \tan A - \sin A = n$ , then the value of $m^{2} - n^{2} =$

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a

$4 \sqrt{m n}$

b

$5 {(m n)}^{\frac{1}{3}}$

c

$\sqrt{m n}$

d

${(m n)}^{\frac{1}{3}}$

answer is D.

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Detailed Solution

$\begin{array}{l} m^{2} - n^{2} = (m + n) (m - n) = 4 \tan A . \sin A \\ 4 \sqrt{m n} = \sqrt{(\tan A + \sin A) (\tan A - \sin A)} \\ = 4 \sqrt{\tan^{2} A - \sin^{2} A} \\ = 4 \sqrt{\frac{\sin^{2} A}{\cos^{2} A} - \sin^{2} A} \\ = 4 \sin A . \sqrt{\sec^{2} A - 1} \\ = 4. \sin A . \cos A \end{array}$

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