First slide

Trigonometric Identities

Question

If $\tan θ + \sin θ = m and \tan θ - \sin θ = n,$ then

Moderate

A

$m^{2} - n^{2} = 4 mn$
B

$m^{2} + n^{2} = 4 mn$
C

$m^{2} - n^{2} = m^{2} + n^{2}$
D

$m^{2} - n^{2} = 4 \sqrt{mn}$

Solution

From the given relations,
$m + n = 2 \tan θ, m - n = 2 \sin θ$
thus , $m^{2} - n^{2} = 4 \tan θsin θ - - - - - (i)$
also , $\sqrt{mn} = \sqrt{\tan^{2} θ - \sin^{2} θ} = \sin θtan θ - - - - (ii)$
From Eqs. (i) and (ii), we get $m^{2} - n^{2} = 4 \sqrt{mn}$

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