Question

Evaluate `(1-costheta)(1+sectheta)=tantheta`?

Answer 1

`theta=npi`

Explanation:

`(1-costheta)(1+sectheta)=tantheta`

• Let's distribute the terms:

`1+sectheta-costheta-costhetasectheta=tantheta`

• Let's now use `costheta=1/sectheta`:

`1+1/costheta-costheta-costheta1/costheta=tantheta`

`1+1/costheta-costheta-1=tantheta`

`1/costheta-costheta=tantheta`

• Let's combine the left side:

`1/costheta-costheta(costheta/costheta)=tantheta`

`1/costheta-cos^2theta/costheta=tantheta`

`(1-cos^2theta)/costheta=tantheta`

• We can now use the identity: `sin^2theta+cos^2theta=1=>sin^2theta=1-cos^2theta`

`sin^2theta/costheta=sintheta/costheta`

i.e. `sintheta(1-sintheta)=0`

Therefore, either `sintheta=0` or `1-sintheta=0` i.e. `sintheta=1`

But at `sintheta=1`, we have both `tantheta` and `sectheta` as undefined and hence `theta=npi`.

Answer 2

`t = kpi,` and
`t = pi/2 + 2kpi`

Explanation:

`( 1 - cos t)(1 + 1/(cos t)) = tan t`
`((1 - cos t)(cos t + 1))/cos t = sin t/(cos t)`
`1 - cos^2 t = sin t`
`sin^2 t - sin t = 0`
`sin t( sin t - 1) = 0`
Use unit circle to solve this equation.
a. sin t = 0 --> t = 0, `t= pi`, and `t = 2pi`
b. sin t - 1 = 0 --> sin t = 1 --> `t = pi/2`
General answers:
`t = kpi`
`t = pi/2 + 2kpi`