Question

How do you simplify `Sin (x+(pi/6)) - sin (x-(pi/6))`?

Answer 1

cosx

Explanation:

Using `color(blue)" Addition Formulae"`

sin(A ± B ) = sinAcosB ± cosAsinB

(1) `sin(x+pi/6 ) = sinxcos(pi/6 )+ cosxsin(pi/6)`

(2) `sin(x - pi/6 ) = sinxcos(pi/6) - cosxsin(pi/6)`

Placing expansions from (1) and (2) back into original expression.

`sinxcos(pi/6) +cosxsin(pi/6) - [sinxcos(pi/6) - cosxsin(pi/6)]`

now `sinxcos(pi/6) - sinxcos(pi/6) = 0`

and `cosxsin(pi/6) + cosxsin(pi/6) = 2cosxsin(pi/6)`

`rArr2cosxsin(pi/6) = 2cosx xx 1/2 = cosx`