Is (sin(2θ))/(sin(θ))-(cos(2θ))/(cos(θ))=sec(θ) ?

The answer to whether (sin(2θ))/(sin(θ))-(cos(2θ))/(cos(θ))=sec(θ) is True

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prove (sin(2θ))/(sin(θ))-(cos(2θ))/(cos(θ))=sec(θ)

prove

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

True

Solution steps

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

Manipulating left side

\frac{sin ( 2 θ )}{sin ( θ )} - \frac{cos ( 2 θ )}{cos ( θ )}

Simplify

- \frac{cos ( 2 θ )}{cos ( θ )} + \frac{sin ( 2 θ )}{sin ( θ )} : \frac{- cos ( 2 θ ) sin ( θ ) + sin ( 2 θ ) cos ( θ )}{cos ( θ ) sin ( θ )}

= \frac{- cos ( 2 θ ) sin ( θ ) + cos ( θ ) sin ( 2 θ )}{cos ( θ ) sin ( θ )}

Rewrite using trig identities

= \frac{1}{cos ( θ )}

Rewrite using trig identities

sec (θ)

We showed that the two sides could take the same form

\Rightarrow True

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

\frac{sin ( θ ) + tan ( θ )}{1 + sec ( θ )} = sin (θ)

sec (2 θ) = \frac{sec ^{2} ( θ )}{1 - tan ^{2} ( θ )}

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

tan (θ) + cot (θ) = sec (θ) \cdot csc (θ)

Is (sin(2θ))/(sin(θ))-(cos(2θ))/(cos(θ))=sec(θ) ?
The answer to whether (sin(2θ))/(sin(θ))-(cos(2θ))/(cos(θ))=sec(θ) is True