Is cot^2(x)=csc^2(x)-csc^2(x)sin^2(x) ?

The answer to whether cot^2(x)=csc^2(x)-csc^2(x)sin^2(x) is True

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prove cot^2(x)=csc^2(x)-csc^2(x)sin^2(x)

prove

cot^{2} (x) = csc^{2} (x) - csc^{2} (x) sin^{2} (x)

True

Solution steps

cot^{2} (x) = csc^{2} (x) - csc^{2} (x) sin^{2} (x)

Manipulating left side

cot^{2} (x)

Express with sin, cos

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

Rewrite using trig identities

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

Manipulating right side

csc^{2} (x) - csc^{2} (x) sin^{2} (x)

Express with sin, cos

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

We showed that the two sides could take the same form

\Rightarrow True

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

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

sin^{2} (x) + cos^{2} (x) = cos (x) sec (x)

sec (y) + tan (y) = \frac{cos ( y )}{1 - sin ( y )}

(csc (x) + cot (x)) (sec (x) - 1) = tan (x)

Is cot^2(x)=csc^2(x)-csc^2(x)sin^2(x) ?
The answer to whether cot^2(x)=csc^2(x)-csc^2(x)sin^2(x) is True