Question

Simplify `cot theta - tan theta`?

Answer 1

`cot theta - tan theta -= 2cot(2theta)`

Explanation:

We can write the expression as:

`cot theta - tan theta -= costheta/sintheta - sintheta/costheta`

`" " = (costheta costheta - sintheta sintheta)/(sinthetacostheta`

`" " = (cos^2theta - sin^2theta)/(sinthetacostheta`

And using the identities:

`sin2A-= 2sinAcosA`
`cos2A-= cos^2A-sin^2A`

We have:

`cot theta - tan theta -= (cos2theta)/(1/2sin2theta`

`" " = 2cot(2theta)`

We can verify the graphically:

`cot theta - tan theta`
graph{cot x - tan x [-10, 10, -5, 5]}

`2cot(2theta)`
graph{2cot(2x) [-10, 10, -5, 5]}