Is (1-csc^2(t))/(csc^2(t))=(-1)/(sec^2(t)) ?

The answer to whether (1-csc^2(t))/(csc^2(t))=(-1)/(sec^2(t)) is True

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prove (1-csc^2(t))/(csc^2(t))=(-1)/(sec^2(t))

prove

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

True

Solution steps

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

Manipulating left side

\frac{1 - csc ^{2} ( t )}{csc ^{2} ( t )}

Express with sin, cos

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

Rewrite using trig identities

= - cos^{2} (t)

Manipulating right side

\frac{- 1}{sec ^{2} ( t )}

Express with sin, cos

= - cos^{2} (t)

We showed that the two sides could take the same form

\Rightarrow True

cot (θ) = cos (θ) csc (θ)

sech^{2} (x) = 1 - tanh^{2} (x)

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

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

sin (\frac{x}{3}) cos (\frac{x}{3}) = \frac{1}{2} sin (\frac{2 x}{3})

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