Is csc(x)+cot(x)=cot(x/2) ?

The answer to whether csc(x)+cot(x)=cot(x/2) is True

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prove csc(x)+cot(x)=cot(x/2)

prove

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

True

Solution steps

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

Let:

u = \frac{x}{2}

csc (2 u) + cot (2 u) = cot (u)

Prove

csc (2 u) + cot (2 u) = cot (u) :

True

csc (2 u) + cot (2 u) = cot (u)

Manipulating left side

csc (2 u) + cot (2 u)

Express with sin, cos

cot (2 u) + csc (2 u)

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Apply rule

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

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

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

= \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

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

We showed that the two sides could take the same form

\Rightarrow True

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

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

3 sin (9 π - x) = 3 sin (x)

sin^{2} (x) = sin (x) sin (x)

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

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