Is (sec(θ)csc(θ))/2 =csc(2θ) ?

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

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prove (sec(θ)csc(θ))/2 =csc(2θ)

prove

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

True

Solution steps

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

Manipulating right 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}

= \frac{sec ( θ ) csc ( θ )}{2}

We showed that the two sides could take the same form

\Rightarrow True

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

\frac{1 + cos ( θ )}{csc ( θ )} = sin (θ) + cos (θ)

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

sin (2 θ) = 2 cos (θ) \cdot sin (θ)

\frac{1 + cos ( a )}{sin ( a )} = cot (\frac{a}{2})

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