$ 1 + s i n θ 1 + 1 − s i n θ 1 = $, 2 cos 2 θ, − 2 cos 2 θ, 2 tan 2 θ, 2 sec 2 θ

1 + sin θ 1 + 1 − sin θ 1 = ( 1 + sin θ ) ( 1 − sin θ ) 1 − sin θ + 1 + sin θ = 1 − sin 2 θ 2 = cos 2 θ 2 = 2 sec 2 θ

$ 1 + s i n θ 1 + 1 − s i n θ 1 = $

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Question: Answered & Verified by Expert

$$
\frac{1}{1+\sin \theta}+\frac{1}{1-\sin \theta}=
$$

MathematicsTrigonometric Ratios & IdentitiesAP EAMCETAP EAMCET 2022 (06 Jul Shift 2)

Options:

A $2 \cos ^2 \theta$
B $-2 \cos ^2 \theta$
C $2 \tan ^2 \theta$
D $2 \sec ^2 \theta$

Solution:

1392 Upvotes Verified Answer

The correct answer is: $2 \sec ^2 \theta$

$\begin{aligned} & \frac{1}{1+\sin \theta}+\frac{1}{1-\sin \theta} \\ & =\frac{1-\sin \theta+1+\sin \theta}{(1+\sin \theta)(1-\sin \theta)}=\frac{2}{1-\sin ^2 \theta}=\frac{2}{\cos ^2 \theta} \\ & =2 \sec ^2 \theta\end{aligned}$

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