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

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

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

prove

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

True

Solution steps

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

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 )}

Simplify

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

Multiply

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

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

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 2 = 2

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

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

Apply the fraction rule:

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

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

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

cos (θ + \frac{π}{2}) = - sin (θ)

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

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

csch^{2} (x) = coth^{2} (x) - 1

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