Is tan^2(-θ)+1=sec^2(θ) ?

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

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prove tan^2(-θ)+1=sec^2(θ)

prove

tan^{2} (- θ) + 1 = sec^{2} (θ)

True

Solution steps

tan^{2} (- θ) + 1 = sec^{2} (θ)

Manipulating left side

tan^{2} (- θ) + 1

Use the negative angle identity:

tan (- x) = - tan (x)

= 1 + (- tan (θ))^{2}

Simplify

= 1 + tan^{2} (θ)

Rewrite using trig identities

1 + tan^{2} (θ)

Use the Pythagorean identity:

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

= sec^{2} (θ)

= sec^{2} (θ)

We showed that the two sides could take the same form

\Rightarrow True

\frac{1 + csc ( β )}{cot ( β ) + cos ( β )} = sec (β)

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

sin (θ - \frac{π}{2}) = cos (π + θ)

\frac{1}{cot ( θ )} = \frac{sec ^{2} ( θ ) - 1}{tan ( θ )}

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