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

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

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

prove

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

True

Solution steps

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

Manipulating right side

cos^{2} (x) + (cot (x) cos (x))^{2}

Express with sin, cos

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

Factor

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

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

Rewrite using trig identities

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

Rewrite using trig identities

cot^{2} (x)

We showed that the two sides could take the same form

\Rightarrow True

sin (- x) + cos (- x) = - sin (x) + cos (x)

cot (θ) sec (θ) sin (θ) = 1

tan (π + θ) = tan (θ)

csc (θ) = cot (θ) sec (θ)

\frac{1 - tan ( x )}{sin ( x )} = csc (x) - sec (x)

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