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

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

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

prove

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

True

Solution steps

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

Manipulating left side

csc (A)

Express with sin, cos

csc (A)

Use the basic trigonometric identity:

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

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

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

Rewrite using trig identities

\frac{1}{sin ( A )}

Use the Double Angle identity:

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

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

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

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

Expand

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

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

Apply the fraction rule:

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

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

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

Cancel

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

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

Cancel the common factor:

sin (A)

= 2 sin (A)

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

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

Manipulating right side

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

Rewrite using trig identities

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

Use the Double Angle identity:

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

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

Cancel

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

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

Cancel the common factor:

cos (A)

= 2 sin (A)

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

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

We showed that the two sides could take the same form

\Rightarrow True

sin (θ) csc (θ) + cot (θ) sin (θ) = 1 + cos (θ)

tan (x) cot (x) = csc (x) sec (x)

sin^{2} (θ) = 1 + cos^{2} (θ)

csc (θ) + cos^{2} (θ) = csc (θ)

sin (\frac{π}{2}) = \frac{4}{2}

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