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$\theta$ = ...

If sin $\theta$ = $\frac{m - n}{m + n}$ ; Find the value ...

If sin $\theta$ = $\frac{m - n}{m + n}$ ; Find the value of 1 + tan2 $\theta$

A.

$\frac{(m^2 + n^2)}{m + n}$

B.

$\frac{(m^2 + n^2 + 2mn)}{4mn}$

C.

$\frac{2(m^2 + n^2 + mn)}{m + n}$

D.

$\frac{(m^2 + n^2 + mn)}{m + n}$

View answer & explanation

Correct answer is B

$(m + n)^{2} = (m - n)^{2} + x^{2}$

$m^{2} + 2mn + n^{2} = m^{2} - 2mn + n^{2} + x^{2}$

$x^{2} = 4mn$

$x = \sqrt{4mn} = 2\sqrt{mn}$

1 + tan2 $\theta$ = sec2 $\theta$

= $\frac{1}{cos^2\theta}$

$\cos \theta = \frac{2\sqrt{mn}}{(m + n)}$

$\frac{1}{\cos \theta} = \frac{(m + n)}{2\sqrt{mn}}$

$\sec^{2} \theta = \frac{(m + n)^{2}}{4mn}$

= $\frac{(m^2 + n^2 + 2mn)}{4mn}$

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