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

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

Popular >

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

prove

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

True

Solution steps

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

Manipulating left side

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

Express with sin, cos

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

Rewrite using trig identities

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

Manipulating right side

2 tan (x) sec (x)

Express with sin, cos

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

We showed that the two sides could take the same form

\Rightarrow True

\frac{1 + cos ( t )}{1 - cos ( t )} = (csc (t) + cot (t))^{2}

sec^{2} (α) + csc^{2} (α) = \frac{1}{sin ^{2} ( α ) cos ^{2} ( α )}

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

tan (x) = sin (x) sec (x)

(1 + cot^{2} (θ)) (1 - cos^{2} (θ)) = 1

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