Is csc^2(x)(1-sin^2(x))=cot^2(x) ?

The answer to whether csc^2(x)(1-sin^2(x))=cot^2(x) is True

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prove csc^2(x)(1-sin^2(x))=cot^2(x)

prove

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

True

Solution steps

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

Manipulating right 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

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

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

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

Use the basic trigonometric identity:

\frac{1}{sin ( x )} = csc (x)

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

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

We showed that the two sides could take the same form

\Rightarrow True

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