Is sin(2x)=2(cos(x))/(csc(x)) ?

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

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prove sin(2x)=2(cos(x))/(csc(x))

prove

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

True

Solution steps

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

Manipulating right side

2 \frac{cos ( x )}{csc ( x )}

Express with sin, cos

2 \cdot \frac{cos ( x )}{csc ( x )}

Use the basic trigonometric identity:

csc (x) = \frac{1}{sin ( x )}

= 2 \cdot \frac{cos ( x )}{\frac{1}{s i n ( x )}}

Simplify

2 \cdot \frac{cos ( x )}{\frac{1}{s i n ( x )}} : 2 cos (x) sin (x)

2 \cdot \frac{cos ( x )}{\frac{1}{s i n ( x )}}

Multiply fractions:

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

= \frac{cos ( x ) \cdot 2}{\frac{1}{s i n ( x )}}

Apply the fraction rule:

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

= \frac{cos ( x ) \cdot 2 sin ( x )}{1}

Apply rule

\frac{a}{1} = a

= 2 cos (x) sin (x)

= 2 cos (x) sin (x)

= 2 cos (x) sin (x)

Rewrite using trig identities

2 cos (x) sin (x)

Use the Double Angle identity:

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

= sin (2 x)

= sin (2 x)

We showed that the two sides could take the same form

\Rightarrow True

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