Is 1/(2csc(2x))=cos^2(x)tan(x) ?

The answer to whether 1/(2csc(2x))=cos^2(x)tan(x) is True

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prove 1/(2csc(2x))=cos^2(x)tan(x)

prove

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

True

Solution steps

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

Manipulating right side

cos^{2} (x) tan (x)

Express with sin, cos

cos^{2} (x) tan (x)

Use the basic trigonometric identity:

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

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

Simplify

cos^{2} (x) \frac{sin ( x )}{cos ( x )} : sin (x) cos (x)

cos^{2} (x) \frac{sin ( x )}{cos ( x )}

Multiply fractions:

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

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

Cancel the common factor:

cos (x)

= sin (x) cos (x)

= sin (x) cos (x)

= cos (x) sin (x)

Rewrite using trig identities

cos (x) sin (x)

Use the Double Angle identity:

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

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

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

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

Rewrite using trig identities

Use the basic trigonometric identity:

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

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

Simplify

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

Apply the fraction rule:

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

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

(sec (5 x) - 1) (sec (5 x) - 1) = tan^{2} (5 x)

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

cot (x) + tan (x) = sec (x) \cdot csc (x)

cot (4 x) = \frac{1 - tan ^{2} ( 2 x )}{2 tan ( 2 x )}

sin (16 x) = 2 sin (8 x) cos (8 x)

Is 1/(2csc(2x))=cos^2(x)tan(x) ?
The answer to whether 1/(2csc(2x))=cos^2(x)tan(x) is True