Is (tan^2(x))/(1+tan^2(x))=sin^2(x) ?

The answer to whether (tan^2(x))/(1+tan^2(x))=sin^2(x) is True

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prove (tan^2(x))/(1+tan^2(x))=sin^2(x)

prove

\frac{tan ^{2} ( x )}{1 + tan ^{2} ( x )} = sin^{2} (x)

True

Solution steps

\frac{tan ^{2} ( x )}{1 + tan ^{2} ( x )} = sin^{2} (x)

Manipulating left side

\frac{tan ^{2} ( x )}{1 + tan ^{2} ( x )}

Express with sin, cos

= \frac{sin ^{2} ( x )}{cos ^{2} ( x ) + sin ^{2} ( x )}

Rewrite using trig identities

= sin^{2} (x)

We showed that the two sides could take the same form

\Rightarrow True

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

tan (\frac{π}{2} - x) sec (x) = csc (x)

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

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

\frac{tan ^{2} ( x )}{sec ( x )} + cos (x) = sec (x)

Is (tan^2(x))/(1+tan^2(x))=sin^2(x) ?
The answer to whether (tan^2(x))/(1+tan^2(x))=sin^2(x) is True