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

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

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

prove

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

True

Solution steps

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

Manipulating left side

\frac{cos ( x )}{sec ( x ) - 1} - \frac{cos ( x )}{sec ( x ) + 1}

Expand

\frac{cos ( x )}{- 1 + sec ( x )} - \frac{cos ( x )}{1 + sec ( x )} : \frac{2 cos ( x )}{sec ^{2} ( x ) - 1}

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

Manipulating right side

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

Rewrite using trig identities

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

We showed that the two sides could take the same form

\Rightarrow True

\frac{cot ( a )}{cos ( a )} = csc (a)

cos (2 θ) = \frac{2 - sec ^{2} ( θ )}{sec ^{2} ( θ )}

cos^{2} (x) tan^{2} (x) + cos^{2} (x) = 1

\frac{csc ( θ ) sin ( θ )}{sec ( θ )} = cos (θ)

tan (x) cot (x) - sin^{2} (x) = cos^{2} (x)

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