Complete the identity.
-The expression $\frac { 1 } { 1 + \sin x } + \frac { 1 } { 1 - \sin x }$
is to be the left hand side of an equation that is an identity. Which one of the following four expressions can be us the right hand side of the equation to complete the identity?
A) $1 - \cos \theta$
B) $2 \sec ^ { 2 } \theta$
C) $\tan ^ { 2 } \theta$
D) $3 \sec \theta$ as

Question

Quizplus · Accepted Answer

To find the correct identity, we simplify the given expression. The expression is $\frac { 1 } { 1 + \sin x } + \frac { 1 } { 1 - \sin x }$. To simplify, find a common denominator and combine the fractions:$$\frac { 1 } { 1 + \sin x } + \frac { 1 } { 1 - \sin x } = \frac{(1 - \sin x) + (1 + \sin x)}{(1 + \sin x)(1 - \sin x)}$$Simplifying the numerator and the denominator gives:$$\frac{2}{1 - \sin^2 x}$$Using the Pythagorean identity, $1 - \sin^2 x = \cos^2 x$, we can rewrite the expression as:$$\frac{2}{\cos^2 x} = 2 \sec^2 x$$Therefore, the correct identity that completes the given expression is $2 \sec^2 x$, which corresponds to choice B.

Complete the Identity 11+sin⁡x+11−sin⁡x\frac { 1 } { 1 + \sin x } + \frac { 1 } { 1 - \sin x }1+sinx1​+1−sinx1​

Complete the Identity $\frac { 1 } { 1 + \sin x } + \frac { 1 } { 1 - \sin x }$