Is (sin(2x))/(sin(x))= 2/(sec(x)) ?

The answer to whether (sin(2x))/(sin(x))= 2/(sec(x)) is True

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prove (sin(2x))/(sin(x))= 2/(sec(x))

prove

\frac{sin ( 2 x )}{sin ( x )} = \frac{2}{sec ( x )}

True

Solution steps

\frac{sin ( 2 x )}{sin ( x )} = \frac{2}{sec ( x )}

Manipulating left side

\frac{sin ( 2 x )}{sin ( x )}

Rewrite using trig identities

\frac{sin ( 2 x )}{sin ( x )}

Use the Double Angle identity:

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

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

Cancel the common factor:

sin (x)

= 2 cos (x)

= 2 cos (x)

Rewrite using trig identities

Use the basic trigonometric identity:

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

2 \cdot \frac{1}{sec ( x )}

Simplify

2 \cdot \frac{1}{sec ( x )}

Multiply fractions:

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

= \frac{1 \cdot 2}{sec ( x )}

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2}{sec ( x )}

\frac{2}{sec ( x )}

\frac{2}{sec ( x )}

We showed that the two sides could take the same form

\Rightarrow True

1 - tanh^{2} (z) = sech^{2} (z)

csc (s) = cot (s) sec (s)

csc^{4} (x) - cot^{4} (x) = 2 sec^{2} (x) - 1

\frac{2}{1 + cos ( 2 x )} = sec^{2} (x)

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

Is (sin(2x))/(sin(x))= 2/(sec(x)) ?
The answer to whether (sin(2x))/(sin(x))= 2/(sec(x)) is True