Question

Solve tan x - square root 1-2tan^2x=0 given that 0 degrees < x<360degrees

Answer 1

x = 30 or x = 210

Explanation:

Given equation is:

`\[\tan x - \sqrt{1 - 2*\tan ^(2) x} = 0\]`

Simplifying,

`\[\tan x = \sqrt{1 - 2*\tan ^(2) x}\]`

Squaring both sides,

`\[\tan ^(2) x = 1 - 2*\tan ^(2) x\]`

=>
`\[\tan ^(2) x + 2*\tan ^(2) x = 1\]`

=>
`\[\3*\tan ^(2) x= 1\]`

=>
`\[\tan x= \pm (1)/(√(3))\]`

Solving for x which satisfies the above equality and also 0 < x < 360,

x = 30 or x = 210

Answer 2

x=30° or 210°

Explanation:

The given equation is:

`tanx-\sqrt{1-2tan^(2)x}=0`

Taking the second term to RHS we get

`tanx=\sqrt{1-2tan^(2)x }`

Squaring both sides of the equation,we get

`tan^(2)x=1-2tan^(2)x`

`3tan^(2)x=1`

`tan^(2)x=(1)/(3)`

∴tanx=±
`(1)/(√(3) )`

But tanx cannot be negative as RHS in the given equation will be positive always. Hence tanx=
`(1)/(√(3) )`

∴ x=30° or x=180°+30°=210° (As tan is positive in first and third quadrant)