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

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

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

prove

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

True

Solution steps

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

Manipulating left side

csc (2 x)

Express with sin, cos

csc (2 x)

Use the basic trigonometric identity:

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

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

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

Rewrite using trig identities

\frac{1}{sin ( 2 x )}

Use the Double Angle identity:

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

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

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

Rewrite using trig identities

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

\frac{1}{2 \cdot \frac{1}{c s c ( x )} \cdot \frac{1}{s e c ( x )}}

Simplify

\frac{1}{2 \cdot \frac{1}{c s c ( x )} \cdot \frac{1}{s e c ( x )}}

Multiply

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

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

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 1 \cdot 2 = 2

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

= \frac{1}{\frac{2}{c s c ( x ) s e c ( x )}}

Apply the fraction rule:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

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

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

p ro v e sec (2 x) = \frac{( sec ^{2} ( x ) )}{2 - sec ^{2} ( x )}

p ro v e sin (x) = 0

p ro v e sin (x) = 1

p ro v e sin (x + 3 0^{\circ}) + 3 cos (x + 3 0^{\circ}) = 2 cos (x)

p ro v e \frac{8 csc ( - x )}{sec ( - x )} = - 8 cot (x)

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