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

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

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

prove

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

True

Solution steps

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

Manipulating left side

tan^{2} (t)

Express with sin, cos

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

Rewrite using trig identities

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

Expand

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

(2 sin (x) cos (x))^{4} = sin^{4} (2 x)

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

csc (θ) + sec (θ) = tan (θ) + 1

\frac{cos ( t ) + 1}{sin ( t ) + sec ( t )} = \frac{tan ( t )}{sec ( t ) - 1}

\frac{cot ( θ ) - tan ( θ )}{cot ( θ ) + tan ( θ )} = cos (2 θ)

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