Question

If x = 30°, without using tables, verify that : `sin 2x = (2 tan x)/(1 + tan^2 x)`.

Answer 1

Given: x = 30°, verify `sin 2x = (2 tan x)/(1 + tan^2 x)`.

Step-wise calculation:

1. Start from the right-hand side:

`(2 tan x)/(1 + tan^2 x) = (2((sin x)/(cos x)))/(1 + ((sin^2 x)/(cos^2 x))`

2. Combine the denominator:

`1 + (sin^2 x/cos^2 x) = (cos^2 x + sin^2 x)/(cos^2 x)`   

= `1/(cos^2 x)`   ...(Since sin² x + cos² x = 1)

3. So, the RHS becomes

`((2 sin x)/(cos x)) ÷ (1/(cos^2 x))` 

= `((2 sin x)/(cos x)) xx cos^2 x` 

= 2 sin x cos x

4. Use the double-angle identity for sine:

2 sin x cos x = sin 2x

Hence, algebraically for all x where tan is defined `(2 tan x)/(1 + tan^2 x) = sin 2x`.

Numeric check for x = 30°

sin 2x = sin 60°

`sin 2x = sqrt(3)/2`

`tan 30^circ = 1/sqrt(3)` 

So, RHS = `(2 xx 1/sqrt(3))/(1 + 1/3)`

= `(2/sqrt(3))/(4/3)`

= `2/sqrt(3) xx 3/4` 

= `3/(2sqrt(3))` 

= `sqrt(3)/2`

Both algebraic simplification and the numeric check for x = 30° give `sin 2x = (2 tan x)/(1 + tan^2 x)`. Identity verified.