Is (cos(2X))/(sin(X))+(sin(2X))/(cos(X))=csc(X) ?

The answer to whether (cos(2X))/(sin(X))+(sin(2X))/(cos(X))=csc(X) is True

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prove (cos(2X))/(sin(X))+(sin(2X))/(cos(X))=csc(X)

prove

\frac{cos ( 2 X )}{sin ( X )} + \frac{sin ( 2 X )}{cos ( X )} = csc (X)

True

Solution steps

\frac{cos ( 2 X )}{sin ( X )} + \frac{sin ( 2 X )}{cos ( X )} = csc (X)

Manipulating left side

\frac{cos ( 2 X )}{sin ( X )} + \frac{sin ( 2 X )}{cos ( X )}

Rewrite using trig identities

\frac{cos ( 2 X )}{sin ( X )} + \frac{sin ( 2 X )}{cos ( X )}

Use the Double Angle identity:

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

= \frac{cos ( 2 X )}{sin ( X )} + \frac{2 sin ( X ) cos ( X )}{cos ( X )}

Cancel

\frac{2 sin ( X ) cos ( X )}{cos ( X )} : 2 sin (X)

\frac{2 sin ( X ) cos ( X )}{cos ( X )}

Cancel the common factor:

cos (X)

= 2 sin (X)

= \frac{cos ( 2 X )}{sin ( X )} + 2 sin (X)

= \frac{cos ( 2 X )}{sin ( X )} + 2 sin (X)

Manipulating right side

csc (X)

Express with sin, cos

csc (X)

Use the basic trigonometric identity:

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

= \frac{1}{sin ( X )}

= \frac{1}{sin ( X )}

Rewrite using trig identities

\frac{1}{sin ( X )}

Use the Double Angle identity:

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

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

= \frac{cos ( 2 X ) + 2 sin ^{2} ( X )}{sin ( X )}

= \frac{cos ( 2 X ) + 2 sin ^{2} ( X )}{sin ( X )}

Expand

\frac{cos ( 2 X ) + 2 sin ^{2} ( X )}{sin ( X )} : \frac{cos ( 2 X )}{sin ( X )} + 2 sin (X)

\frac{cos ( 2 X ) + 2 sin ^{2} ( X )}{sin ( X )}

Apply the fraction rule:

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

\frac{cos ( 2 X ) + 2 sin ^{2} ( X )}{sin ( X )} = \frac{cos ( 2 X )}{sin ( X )} + \frac{2 sin ^{2} ( X )}{sin ( X )}

= \frac{cos ( 2 X )}{sin ( X )} + \frac{2 sin ^{2} ( X )}{sin ( X )}

Cancel

\frac{2 sin ^{2} ( X )}{sin ( X )} : 2 sin (X)

\frac{2 sin ^{2} ( X )}{sin ( X )}

Cancel the common factor:

sin (X)

= 2 sin (X)

= \frac{cos ( 2 X )}{sin ( X )} + 2 sin (X)

= \frac{cos ( 2 X )}{sin ( X )} + 2 sin (X)

We showed that the two sides could take the same form

\Rightarrow True

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

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

tan^{5} (x) = tan^{3} (x) sec^{2} (x) - tan^{3} (x)

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

tan^{2} (x) (csc^{2} (x) - 1) = 1

Is (cos(2X))/(sin(X))+(sin(2X))/(cos(X))=csc(X) ?
The answer to whether (cos(2X))/(sin(X))+(sin(2X))/(cos(X))=csc(X) is True