Is tan(x)-(sec^2(x))/(tan(x))=-cot(x) ?

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

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prove tan(x)-(sec^2(x))/(tan(x))=-cot(x)

prove

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

True

Solution steps

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

Manipulating left side

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

Express with sin, cos

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

Rewrite using trig identities

= - \frac{cos ( x )}{sin ( x )}

Use the basic trigonometric identity:

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

= - cot (x)

We showed that the two sides could take the same form

\Rightarrow True

\frac{tan ^{2} ( θ ) + 1}{tan ^{2} ( θ )} = csc^{2} (θ)

sec^{2} (x) csc^{2} (x) - sec^{2} (x) = csc^{2} (x)

\frac{8 sin ^{2} ( t )}{tan ^{2} ( t )} = 8 cos^{2} (t)

\frac{sin ( θ ) + sin ( 3 θ )}{2 cos ( θ )} = sin (2 θ)

tan^{2} (θ) = \frac{1}{cos ^{2} ( θ )} - 1

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