Is cot(x)=(1+cos(2x))/(sin(2x)) ?

The answer to whether cot(x)=(1+cos(2x))/(sin(2x)) is True

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prove cot(x)=(1+cos(2x))/(sin(2x))

prove

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

True

Solution steps

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

Manipulating right side

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

Rewrite using trig identities

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

Use the Double Angle identity:

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

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

Use the Double Angle identity:

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

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

Simplify

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

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

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

1 + 2 cos^{2} (x) - 1

Group like terms

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

1 - 1 = 0

= 2 cos^{2} (x)

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

Divide the numbers:

\frac{2}{2} = 1

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

Cancel the common factor:

cos (x)

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

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

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

Use the basic trigonometric identity:

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

= cot (x)

We showed that the two sides could take the same form

\Rightarrow True

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

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

cot (θ) sin (θ) = cos (θ)

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

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

Is cot(x)=(1+cos(2x))/(sin(2x)) ?
The answer to whether cot(x)=(1+cos(2x))/(sin(2x)) is True