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

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

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prove (1-sin^2(A))/(1-cos^2(A))=cot^2(A)

prove

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

True

Solution steps

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

Manipulating right side

cot^{2} (A)

Express with sin, cos

cot^{2} (A)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Apply exponent rule:

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

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

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

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

Rewrite using trig identities

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

Use the Double Angle identity:

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

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

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

Use the Pythagorean identity:

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

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

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

Use the Double Angle identity:

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

= \frac{cos ^{2} ( A ) - sin ^{2} ( A ) + 1 - cos ^{2} ( A )}{1 - cos ^{2} ( A )}

cos^{2} (A) - sin^{2} (A) + 1 - cos^{2} (A) = - sin^{2} (A) + 1

cos^{2} (A) - sin^{2} (A) + 1 - cos^{2} (A)

Group like terms

= cos^{2} (A) - sin^{2} (A) - cos^{2} (A) + 1

Add similar elements:

cos^{2} (A) - cos^{2} (A) = 0

= - sin^{2} (A) + 1

= \frac{- sin ^{2} ( A ) + 1}{1 - cos ^{2} ( A )}

= \frac{- sin ^{2} ( A ) + 1}{1 - cos ^{2} ( A )}

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

We showed that the two sides could take the same form

\Rightarrow True

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