Question

Solve `2tan^2xsinx-tan^2x=0`, if `x in[0^@,360^@]`?

Answer 1

`x in(0^o,30^o,150^o,180^o,360^o)`

Explanation:

`2tan^2xsinx-tan^2x=0`

`2sin^2x/cos^2xsinx-sin^2x/cos^2x=0`

Multiply through by `cos^2x`:

`2sin^2xsinx-sin^2x=0`

`2sin^3x-sin^2x=0`

`sin^2x(2sinx-1)=0`

So either:

`sin^2x=0` or `2sinx-1=0`

Solving for `sin^2x=0`

`-> sin(x)=0-> x=sin^(-1)(0)`

In the interval of `0^o` to `360^o` 3 possible values satisfy this, namely: `0^o,180^o` and `360^o`.

Solving:

`2sinx-1=0-> x=sin^(-1)(1/2)`

To values satisfy this in the interval, namely:

`x=30^o` and `x=180^o-30^o=150^o`

So `x in(0^o,30^o,150^o,180^o,360^o)`