Is (1-sin^2(x))(1-sec^2(x))=-sin^2(x) ?

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

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prove (1-sin^2(x))(1-sec^2(x))=-sin^2(x)

prove

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

True

Solution steps

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

Manipulating left side

(1 - sin^{2} (x)) (1 - sec^{2} (x))

Expand

(1 - sec^{2} (x)) (1 - sin^{2} (x)) : 1 - sin^{2} (x) - sec^{2} (x) + sec^{2} (x) sin^{2} (x)

= 1 - sec^{2} (x) - sin^{2} (x) + sec^{2} (x) sin^{2} (x)

Rewrite using trig identities

= - sin^{2} (x) + sec^{2} (x) sin^{2} (x) - tan^{2} (x)

Express with sin, cos

= - sin^{2} (x)

We showed that the two sides could take the same form

\Rightarrow True

\frac{1}{csc ( x ) + 1} + \frac{1}{sin ( x ) + 1} = 1

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

cos (θ) cot (θ) + sin (θ) = csc (θ)

cos (2 u) = cos^{2} (u) - sin^{2} (u)

(csc (θ) + 1) (csc (θ) - 1) = cot^{2} (θ)

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