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If $\tan A + \sin A = m$ and $\tan A - \sin A = n$ then $\frac{{(m^{2} - n^{2})}^{2}}{mn}$ is equal to

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a

4

b

3

c

16

d

9

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detailed solution

Correct option is C

Since,tan⁡A+sin⁡A=mand tan⁡A−sin⁡A=n∴ m+n=2tan⁡Aand m−n=2sin⁡AAlso, mn=(tan⁡A+sin⁡A)(tan⁡A−sin⁡A) =tan2⁡A−sin2⁡ANow, m2−n22mn=(m+n)2(m−n)2mn=(2tan⁡A)2(2sin⁡A)2tan2⁡A−sin2⁡A=16tan2⁡Asin2⁡Asin2⁡Atan2⁡A=16

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