Is sec^2(-θ)-1=tan^2(θ) ?

The answer to whether sec^2(-θ)-1=tan^2(θ) is True

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prove sec^2(-θ)-1=tan^2(θ)

prove

sec^{2} (- θ) - 1 = tan^{2} (θ)

True

Solution steps

sec^{2} (- θ) - 1 = tan^{2} (θ)

Manipulating left side

sec^{2} (- θ) - 1

Use the negative angle identity:

sec (- x) = sec (x)

= - 1 + sec^{2} (θ)

Manipulating right side

tan^{2} (θ)

Rewrite using trig identities

tan^{2} (θ)

Use the Pythagorean identity:

tan^{2} (x) + 1 = sec^{2} (x)

tan^{2} (x) = sec^{2} (x) - 1

= sec^{2} (θ) - 1

= sec^{2} (θ) - 1

We showed that the two sides could take the same form

\Rightarrow True

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

sin (- θ) sec (- θ) cot (- θ) = 1

cos^{4} (θ) - sin^{4} (θ) = 1 - 2 sin^{2} (θ)

cot (x) sin (x) sec^{2} (x) = sec (x)

tan (2 x) = \frac{sin ( 2 x )}{cos ( 2 x )}