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

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

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

prove

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

True

Solution steps

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

Let:

u = \frac{x}{2}

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

Prove

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

True

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

Manipulating left side

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

Rewrite using trig identities

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

Use the Double Angle identity:

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

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

Use the Double Angle identity:

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

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

Simplify

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

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

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

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

Group like terms

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

1 - 1 = 0

= 2 cos^{2} (u)

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

Divide the numbers:

\frac{2}{2} = 1

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

Cancel the common factor:

cos (u)

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

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

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

Use the basic trigonometric identity:

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

= cot (u)

We showed that the two sides could take the same form

\Rightarrow True

Therefore

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

We showed that the two sides could take the same form

\Rightarrow True

(sin (x) - 1) (tan (x) + sec (x)) = - cos (x)

cos (a - b) - cos (a + b) = 2 sin (a) sin (b)

tan (x) + cot (2 x) = csc (2 x)

tan (θ) sin (θ) = sec (θ) - cos (θ)

\frac{sec ( θ )}{tan ( θ )} = csc (θ)

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