Is (2cot(x))/(csc^2(x))=sin(2x) ?

The answer to whether (2cot(x))/(csc^2(x))=sin(2x) is True

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prove (2cot(x))/(csc^2(x))=sin(2x)

prove

\frac{2 cot ( x )}{csc ^{2} ( x )} = sin (2 x)

True

Solution steps

\frac{2 cot ( x )}{csc ^{2} ( x )} = sin (2 x)

Manipulating left side

\frac{2 cot ( x )}{csc ^{2} ( x )}

Express with sin, cos

\frac{2 cot ( x )}{csc ^{2} ( x )}

Use the basic trigonometric identity:

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

= \frac{2 \cdot \frac{c o s ( x )}{s i n ( x )}}{csc ^{2} ( x )}

Use the basic trigonometric identity:

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

= \frac{2 \cdot \frac{c o s ( x )}{s i n ( x )}}{( \frac{1}{s i n ( x )} ) ^{2}}

Simplify

\frac{2 \cdot \frac{c o s ( x )}{s i n ( x )}}{( \frac{1}{s i n ( x )} ) ^{2}} : 2 cos (x) sin (x)

\frac{2 \cdot \frac{c o s ( x )}{s i n ( x )}}{( \frac{1}{s i n ( x )} ) ^{2}}

(\frac{1}{sin ( x )})^{2} = \frac{1}{sin ^{2} ( x )}

(\frac{1}{sin ( x )})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{1 ^{2}}{sin ^{2} ( x )}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{1}{sin ^{2} ( x )}

= \frac{2 \cdot \frac{c o s ( x )}{s i n ( x )}}{\frac{1}{s i n ^{2} ( x )}}

Multiply

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

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

Multiply fractions:

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

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

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

Divide fractions:

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

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

Refine

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

Cancel the common factor:

sin (x)

= 2 cos (x) sin (x)

= 2 cos (x) sin (x)

= 2 cos (x) sin (x)

Rewrite using trig identities

2 cos (x) sin (x)

Use the Double Angle identity:

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

= sin (2 x)

= sin (2 x)

We showed that the two sides could take the same form

\Rightarrow True

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

tan (x) - cot (x) = - 2 cot (2 x)

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

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

cos^{4} (x) - sin^{4} (x) = 2 cos^{2} (x) - 1

Is (2cot(x))/(csc^2(x))=sin(2x) ?
The answer to whether (2cot(x))/(csc^2(x))=sin(2x) is True