Is (cos(A))/(1+sin(A))+(cos(A))/(1-sin(A))=2sec(A) ?

The answer to whether (cos(A))/(1+sin(A))+(cos(A))/(1-sin(A))=2sec(A) is True

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prove (cos(A))/(1+sin(A))+(cos(A))/(1-sin(A))=2sec(A)

prove

\frac{cos ( A )}{1 + sin ( A )} + \frac{cos ( A )}{1 - sin ( A )} = 2 sec (A)

True

Solution steps

\frac{cos ( A )}{1 + sin ( A )} + \frac{cos ( A )}{1 - sin ( A )} = 2 sec (A)

Manipulating left side

\frac{cos ( A )}{1 + sin ( A )} + \frac{cos ( A )}{1 - sin ( A )}

Rewrite using trig identities

= \frac{cos ( A )}{sin ( A ) + 1} + \frac{sin ( A ) + 1}{cos ( A )}

Simplify

\frac{1 + sin ( A )}{cos ( A )} + \frac{cos ( A )}{1 + sin ( A )} : \frac{( 1 + sin ( A ) ) ^{2} + cos ^{2} ( A )}{cos ( A ) ( sin ( A ) + 1 )}

= \frac{( 1 + sin ( A ) ) ^{2} + cos ^{2} ( A )}{( 1 + sin ( A ) ) cos ( A )}

Rewrite using trig identities

= \frac{2}{cos ( A )}

Rewrite using trig identities

2 sec (A)

We showed that the two sides could take the same form

\Rightarrow True

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Is (cos(A))/(1+sin(A))+(cos(A))/(1-sin(A))=2sec(A) ?
The answer to whether (cos(A))/(1+sin(A))+(cos(A))/(1-sin(A))=2sec(A) is True