Question

What is `cos(theta/2)-2cot(theta/2)` in terms of trigonometric functions of a unit `theta`?

Answer 1

`{sintheta-2sqrt(2(1+costheta))}/sqrt(2(1-costheta)`.

Explanation:

`cos(theta/2)-2cot(theta/2)=cos(theta/2)-2*cos(theta/2)/sin(theta/2)`

`={cos(theta/2)sin(theta/2)-2cos(theta/2)}/sin(theta/2)`.

Multiplying by `2` in `Nr. and Dr.`, we get,

the Exp.`={2cos(theta/2)sin(theta/2)-4cos(theta/2)}/(2sin(theta/2))`.

Here, we use the Identities :

`(i) :2sinxcosx=sin2x; (ii) : 1-cos2x=2sin^2x;`
`(iii) : 1+cos2x=2cos^2x`.

Hence,

`"The Exp.="{sintheta-4sqrt((1+costheta)/2)}/(2sqrt((1-costheta)/2)`

`={sintheta-2sqrt(2(1+costheta))}/sqrt(2(1-costheta)`.

Enjoy Maths.!