Is (1-cos^2(a))/(cos^2(a))=tan^2(a) ?

The answer to whether (1-cos^2(a))/(cos^2(a))=tan^2(a) is True

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prove (1-cos^2(a))/(cos^2(a))=tan^2(a)

prove

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

True

Solution steps

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

Manipulating right side

tan^{2} (a)

Express with sin, cos

tan^{2} (a)

Use the basic trigonometric identity:

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

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

Simplify

(\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 )}

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

p ro v e 1 - sin (r) = cos (r)

p ro v e sin (1 5^{\circ}) = \frac{1 - cos ( 3 0 ^{\circ} )}{2}

p ro v e sin (- 1) = - sin (1)

p ro v e 7 \frac{cot ^{2} ( x )}{csc ( x )} sec^{2} (x) = 7 tan (x) cos (x) csc^{2} (x)

p ro v e cos^{2} (x) = 1 + sin^{2} (x)

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