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

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

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

prove

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

True

Solution steps

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

Manipulating right 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

tan^{2} (u) = \frac{1 - cos ( 2 u )}{1 + cos ( 2 u )}

(1 + sin (θ)) (1 - sin (θ)) = \frac{1}{sec ^{2} ( θ )}

csc^{4} (x) - csc^{2} (x) = cot^{2} (+ cot^{4} (x))

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

cot (x) (cot (x) + tan (x)) = csc^{2} (x)

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