Is tan^2(x)-sin^2(x)=(tan(x)sin(x))^2 ?

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

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prove tan^2(x)-sin^2(x)=(tan(x)sin(x))^2

prove

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

True

Solution steps

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

Manipulating right side

(tan (x) sin (x))^{2}

Simplify

= sin^{2} (x) tan^{2} (x)

Rewrite using trig identities

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

Expand

(1 - cos^{2} (x)) tan^{2} (x) : tan^{2} (x) - tan^{2} (x) cos^{2} (x)

= tan^{2} (x) - cos^{2} (x) tan^{2} (x)

Rewrite using trig identities

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

We showed that the two sides could take the same form

\Rightarrow True

cos^{2} (θ) = sin^{2} (θ) - 1

\frac{sec ( θ )}{sec ( θ ) - 1} = \frac{1}{1 - cos ( θ )}

2 cos (θ) - \frac{cos ( 2 θ )}{cos ( θ )} = sec (θ)

cos (2 \cdot 30) = cos^{2} (30) - sin^{2} (30)

1 + tan^{2} (4 5^{\circ}) = sec^{2} (4 5^{\circ})

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