Is csc(2θ)=((csc(θ)sec(θ)))/2 ?

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

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prove csc(2θ)=((csc(θ)sec(θ)))/2

prove

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

True

Solution steps

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

Manipulating left side

csc (2 θ)

Express with sin, cos

csc (2 θ)

Use the basic trigonometric identity:

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

= \frac{1}{sin ( 2 θ )}

= \frac{1}{sin ( 2 θ )}

Rewrite using trig identities

\frac{1}{sin ( 2 θ )}

Use the Double Angle identity:

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

= \frac{1}{2 sin ( θ ) cos ( θ )}

= \frac{1}{2 sin ( θ ) cos ( θ )}

Rewrite using trig identities

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

Multiply

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

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

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 ( θ ) sec ( θ )}

Multiply the numbers:

1 \cdot 1 \cdot 2 = 2

= \frac{2}{csc ( θ ) sec ( θ )}

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

Apply the fraction rule:

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

= \frac{csc ( θ ) sec ( θ )}{2}

\frac{csc ( θ ) sec ( θ )}{2}

\frac{csc ( θ ) sec ( θ )}{2}

We showed that the two sides could take the same form

\Rightarrow True

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