Is 1/(1+sin(x))+1/(1-sin(x))=2+2tan^2(x) ?

The answer to whether 1/(1+sin(x))+1/(1-sin(x))=2+2tan^2(x) is True

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prove 1/(1+sin(x))+1/(1-sin(x))=2+2tan^2(x)

prove

\frac{1}{1 + sin ( x )} + \frac{1}{1 - sin ( x )} = 2 + 2 tan^{2} (x)

True

Solution steps

\frac{1}{1 + sin ( x )} + \frac{1}{1 - sin ( x )} = 2 + 2 tan^{2} (x)

Manipulating left side

\frac{1}{1 + sin ( x )} + \frac{1}{1 - sin ( x )}

Simplify

\frac{1}{1 + sin ( x )} + \frac{1}{1 - sin ( x )} : \frac{2}{( sin ( x ) + 1 ) ( - sin ( x ) + 1 )}

= \frac{2}{( 1 + sin ( x ) ) ( 1 - sin ( x ) )}

Rewrite using trig identities

= \frac{2}{cos ^{2} ( x )}

Manipulating right side

2 + 2 tan^{2} (x)

Factor

2 + 2 tan^{2} (x) : 2 (1 + tan^{2} (x))

= (1 + tan^{2} (x)) \cdot 2

Express with sin, cos

= \frac{( cos ^{2} ( x ) + sin ^{2} ( x ) ) \cdot 2}{cos ^{2} ( x )}

Rewrite using trig identities

= \frac{2}{cos ^{2} ( x )}

We showed that the two sides could take the same form

\Rightarrow True

tan (a) = \frac{sin ( a )}{cos ( a )}

sec (2 x) = \frac{sec ^{2} ( x )}{1 - tan ^{2} ( x )}

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

\frac{tan ( x ) csc ( x )}{sec ( x )} = 1

\frac{sin ( 3 x ) - sin ( 5 x )}{2 cos ( 4 x )} = - sin (x)

Is 1/(1+sin(x))+1/(1-sin(x))=2+2tan^2(x) ?
The answer to whether 1/(1+sin(x))+1/(1-sin(x))=2+2tan^2(x) is True