Question

Verify the following identities (a) tan 3x =[ tan x(3-tan²x]/(1-3tan²x)?

Answer 1

The proof for a)

Explanation:

We have
`tan(x+2x)=(tan(x)+tan(2x))/(1-tan(x)tan(2x))`
`tan(2x)=2tan(x)/(1-tan(x)^2)`
Putting things together:

`tan(3x)=(tan(x)+2tan(x)/(1-tan^2(x)))/(1-tan^2(x))/(1-tan(x)*(2tan(x)/(1-tan^2(x)))`
Multiplying denominator and numerator by
`1-tan^2(x)`
`tan(3x)=(tan(x)-tan^3(x)+2tan(x))/(1-tan^2(x)-2tan^2(x))`
and this is
`tan(3x)=(tan(x)*(3-tan^2(x)))/(1-3tan^2(x))`