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

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

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

prove

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

True

Solution steps

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

Manipulating right side

cos^{2} (α) + (cot (α) cos (α))^{2}

Express with sin, cos

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

Factor

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

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

Rewrite using trig identities

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

Rewrite using trig identities

cot^{2} (α)

We showed that the two sides could take the same form

\Rightarrow True

cos^{4} (β) - sin^{4} (β) = 2 cos^{2} (β) - 1

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

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

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

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

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