Is (sin(2θ))/(sin(4θ))= 1/2 sec(2θ) ?

The answer to whether (sin(2θ))/(sin(4θ))= 1/2 sec(2θ) is True

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prove (sin(2θ))/(sin(4θ))= 1/2 sec(2θ)

prove

\frac{sin ( 2 θ )}{sin ( 4 θ )} = \frac{1}{2} sec (2 θ)

True

Solution steps

\frac{sin ( 2 θ )}{sin ( 4 θ )} = \frac{1}{2} sec (2 θ)

Manipulating left side

\frac{sin ( 2 θ )}{sin ( 4 θ )}

Rewrite using trig identities

\frac{sin ( 2 θ )}{sin ( 4 θ )}

= \frac{sin ( 2 θ )}{sin ( 2 \cdot 2 θ )}

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

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

Cancel the common factor:

sin (2 θ)

= \frac{1}{2 cos ( 2 θ )}

= \frac{1}{2 cos ( 2 θ )}

Manipulating right side

\frac{1}{2} sec (2 θ)

Express with sin, cos

\frac{1}{2} sec (2 θ)

Use the basic trigonometric identity:

sec (x) = \frac{1}{cos ( x )}

= \frac{1}{2} \cdot \frac{1}{cos ( 2 θ )}

Simplify

\frac{1}{2} \cdot \frac{1}{cos ( 2 θ )} : \frac{1}{2 cos ( 2 θ )}

\frac{1}{2} \cdot \frac{1}{cos ( 2 θ )}

Multiply fractions:

\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}

= \frac{1 \cdot 1}{2 cos ( 2 θ )}

Multiply the numbers:

1 \cdot 1 = 1

= \frac{1}{2 cos ( 2 θ )}

= \frac{1}{2 cos ( 2 θ )}

= \frac{1}{2 cos ( 2 θ )}

We showed that the two sides could take the same form

\Rightarrow True

Is (sin(2θ))/(sin(4θ))= 1/2 sec(2θ) ?
The answer to whether (sin(2θ))/(sin(4θ))= 1/2 sec(2θ) is True