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

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

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

prove

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

True

Solution steps

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

Manipulating left side

\frac{2}{1 - cos ( 2 x )}

Rewrite using trig identities

\frac{2}{1 - cos ( 2 x )}

Use the Double Angle identity:

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

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

Simplify

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

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

Expand

1 - (1 - 2 sin^{2} (x)) : 2 sin^{2} (x)

1 - (1 - 2 sin^{2} (x))

- (1 - 2 sin^{2} (x)) : - 1 + 2 sin^{2} (x)

- (1 - 2 sin^{2} (x))

Distribute parentheses

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

Apply minus-plus rules

- (- a) = a, - (a) = - a

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

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

1 - 1 = 0

= 2 sin^{2} (x)

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

Divide the numbers:

\frac{2}{2} = 1

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

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

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

Rewrite using trig identities

Use the basic trigonometric identity:

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

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

Simplify

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

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

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

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

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

Apply the fraction rule:

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

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

Apply rule

\frac{a}{1} = a

= csc^{2} (x)

csc^{2} (x)

csc^{2} (x)

We showed that the two sides could take the same form

\Rightarrow True

sin^{4} (θ) + cos^{2} (θ) sin^{2} (θ) = sin^{2} (θ)

cos (10 x) - cos (6 x) = - 2 sin (8 x) sin (2 x)

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

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

sec (y) - cos (y) = tan (y) sin (y)

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