Is (sin^2(a))/(1-sin^2(a))=sec^2(a)-1 ?

The answer to whether (sin^2(a))/(1-sin^2(a))=sec^2(a)-1 is True

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prove (sin^2(a))/(1-sin^2(a))=sec^2(a)-1

prove

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

True

Solution steps

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

Manipulating right side

sec^{2} (a) - 1

Express with sin, cos

- 1 + sec^{2} (a)

Use the basic trigonometric identity:

sec (x) = \frac{1}{cos ( x )}

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

Simplify

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

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

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

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

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply:

1 \cdot cos^{2} (a) = cos^{2} (a)

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

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Expand

1 - (1 - sin^{2} (a)) : sin^{2} (a)

1 - (1 - sin^{2} (a))

- (1 - sin^{2} (a)) : - 1 + sin^{2} (a)

- (1 - sin^{2} (a))

Distribute parentheses

= - (1) - (- sin^{2} (a))

Apply minus-plus rules

- (- a) = a, - (a) = - a

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

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

1 - 1 = 0

= sin^{2} (a)

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

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

We showed that the two sides could take the same form

\Rightarrow True

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

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

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

sin (2 θ) + 2 sin (θ) - cos (θ) = 2

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

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