Is cos^2(A)tan^2(A)=sin^2(A) ?

The answer to whether cos^2(A)tan^2(A)=sin^2(A) is True

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prove cos^2(A)tan^2(A)=sin^2(A)

prove

cos^{2} (A) tan^{2} (A) = sin^{2} (A)

True

Solution steps

cos^{2} (A) tan^{2} (A) = sin^{2} (A)

Manipulating left side

cos^{2} (A) tan^{2} (A)

Express with sin, cos

cos^{2} (A) tan^{2} (A)

Use the basic trigonometric identity:

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

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

Simplify

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

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

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

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

Apply exponent rule:

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

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

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

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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

Cancel the common factor:

cos^{2} (A)

= sin^{2} (A)

= sin^{2} (A)

= sin^{2} (A)

We showed that the two sides could take the same form

\Rightarrow True

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Is cos^2(A)tan^2(A)=sin^2(A) ?
The answer to whether cos^2(A)tan^2(A)=sin^2(A) is True