Is cos^2(θ)+cos^2(θ)cot^2(θ)=cot^2(θ) ?

The answer to whether cos^2(θ)+cos^2(θ)cot^2(θ)=cot^2(θ) is True

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prove cos^2(θ)+cos^2(θ)cot^2(θ)=cot^2(θ)

prove

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

True

Solution steps

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

Manipulating left side

cos^{2} (θ) + cos^{2} (θ) cot^{2} (θ)

Express with sin, cos

= \frac{cos ^{4} ( θ ) + cos ^{2} ( θ ) sin ^{2} ( θ )}{sin ^{2} ( θ )}

Factor

\frac{cos ^{4} ( θ ) + cos ^{2} ( θ ) sin ^{2} ( θ )}{sin ^{2} ( θ )} : \frac{cos ^{2} ( θ ) ( cos ^{2} ( θ ) + sin ^{2} ( θ ) )}{sin ^{2} ( θ )}

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

Rewrite using trig identities

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

Rewrite using trig identities

cot^{2} (θ)

We showed that the two sides could take the same form

\Rightarrow True

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

\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 )}

Is cos^2(θ)+cos^2(θ)cot^2(θ)=cot^2(θ) ?
The answer to whether cos^2(θ)+cos^2(θ)cot^2(θ)=cot^2(θ) is True