Is sin(x)+sin(x)tan^2(x)=tan(x)sec(x) ?

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

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prove sin(x)+sin(x)tan^2(x)=tan(x)sec(x)

prove

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

True

Solution steps

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

Manipulating left side

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

Factor

sin (x) + sin (x) tan^{2} (x) : sin (x) (1 + tan^{2} (x))

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

Express with sin, cos

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

Rewrite using trig identities

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

Manipulating right side

tan (x) sec (x)

Express with sin, cos

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

We showed that the two sides could take the same form

\Rightarrow True

sin (θ) (csc (θ) - sin (θ)) = cos^{2} (θ)

sec (y) cos (y) = 1

csc (x) - sin (x) = cos (x) \cdot cot (x)

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

\frac{sec ^{2} ( θ )}{tan ( θ )} = sec (θ) csc (θ)

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