Question

2 tan 30°
II
1 + tan- 300​

Answer 1

Question:

`(2tan30^(\circ))/(1 + tan^2(30^(\circ)))`

Answer:

`(2tan30^(\circ))/(1 + tan^2(30^(\circ)))`
`= sin(60^(\circ))`

Explanation:

Given

`(2tan30^(\circ))/(1 + tan^2(30^(\circ)))`

Required

Simplify

In trigonometry:

`tan(30^(\circ)) = (1)/(√(3))`

So, the expression becomes:

`(2tan30^(\circ))/(1 + tan^2(30^(\circ)))`
`= (2 * (1)/(√(3)))/(1 + ((1)/(√(3)))^2)`

Simplify the denominator

`(2tan30^(\circ))/(1 + tan^2(30^(\circ)))`
`= (2 * (1)/(√(3)))/(1 + (1)/(3))`

`(2tan30^(\circ))/(1 + tan^2(30^(\circ)))`
`= ((2)/(√(3)))/(1 + (1)/(3))`

`(2tan30^(\circ))/(1 + tan^2(30^(\circ)))`
`= ((2)/(√(3)))/( (3+1)/(3))`

`(2tan30^(\circ))/(1 + tan^2(30^(\circ)))`
`= ((2)/(√(3)))/( (4)/(3))`

Express the fraction as:

`(2tan30^(\circ))/(1 + tan^2(30^(\circ)))`
`= (2)/(\sqrt 3) / (4)/(3)`

`(2tan30^(\circ))/(1 + tan^2(30^(\circ)))`
`= (2)/(\sqrt 3) * (3)/(4)`

`(2tan30^(\circ))/(1 + tan^2(30^(\circ)))`
`= (1)/(\sqrt 3) * (3)/(2)`

`(2tan30^(\circ))/(1 + tan^2(30^(\circ)))`
`= (3)/(2\sqrt 3)`

Rationalize

`(2tan30^(\circ))/(1 + tan^2(30^(\circ)))`
`= (3)/(2\sqrt 3) * (√(3))/(√(3))`

`(2tan30^(\circ))/(1 + tan^2(30^(\circ)))`
`= (3√(3))/(2* 3)`

`(2tan30^(\circ))/(1 + tan^2(30^(\circ)))`
`= (√(3))/(2)`

In trigonometry:

`sin(60^(\circ)) = (√(3))/(2)`

Hence:

`(2tan30^(\circ))/(1 + tan^2(30^(\circ)))`
`= sin(60^(\circ))`