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

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

Popular >

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

prove

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

True

Solution steps

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

Manipulating left side

2 tan (x) sec (x)

Express with sin, cos

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

Manipulating right 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 )}{( sin ( x ) + 1 ) ( - sin ( x ) + 1 )}

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

Rewrite using trig identities

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

We showed that the two sides could take the same form

\Rightarrow True

sin^{2} (x) (csc^{2} (x) - 1) = cos^{2} (x)

\frac{1}{sin ( x ) cot ( x )} = \frac{1}{cos ( x )}

\frac{csc ^{2} ( x ) - 1}{csc ^{2} ( x )} = cos^{2} (x)

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

cos (3 α) = 4 cos^{3} (α) - 3 cos (α)

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