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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

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

p ro v e tan^{5} (x) = tan^{3} (x) sec^{2} (x) - tan^{3} (x)

p ro v e \frac{cos ( θ )}{1 - sin ( θ )} = sec (θ) + tan (θ)

p ro v e tan (x) + cot (x) = 2 csc (2 x)

p ro v e sin (\frac{π}{2} - x) = cos (x)

p ro v e tan (2 x) = \frac{( 2 tan ( x ) )}{( 1 - tan ^{2} ( x ) )}

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