Is cot^2(x)=(cos^2(x))/(1-cos^2(x)) ?

The answer to whether cot^2(x)=(cos^2(x))/(1-cos^2(x)) is True

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prove cot^2(x)=(cos^2(x))/(1-cos^2(x))

prove

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

True

Solution steps

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

Manipulating left side

cot^{2} (x)

Express with sin, cos

cot^{2} (x)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

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

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

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

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

3 cos (3 x) = 3 cos (x) - 12 sin^{2} (x) cos (x)

4 sec (y) cos (y) = 4

sin (x) sin (x) = 1

Is cot^2(x)=(cos^2(x))/(1-cos^2(x)) ?
The answer to whether cot^2(x)=(cos^2(x))/(1-cos^2(x)) is True