Question

How do you simplify `sin(x-y)/cos(x+y)-tan(x+y)` to trigonometric functions of x and y?

Answer 1

`(2cosxsiny)/(sinxsiny - cosxcosy)`

Explanation:

Using `color(blue)" Addition formulae "`

• sin( A ± B) = sinAcosB ± cosAsinB

• cos (A ± B ) = cosAcosB ∓ sinAsinB [ note change in signs ]

and trig.identity : `tanx = (sinx)/(cosx)`

using identity for tanx :

`rArr (sin(x-y))/(cos(x+y)) - (sin(x+y))/(cos(x+y))`

common denominator of cos(x+y) , can write as single fraction

`rArr (sin(x-y) - sin(x+y))/(cos(x+y))`

Using 'Addition formulae' to expand numerator/denominator

`color(blue)" expanding numerator "`

sinxcosy - cosxsiny - (sinxcosy +cosxsiny )

= sinxcosy - cosxsiny - sinxcosy - cosxsiny = -2cosxsiny

`color(red)" expanding denominator "`

cos(x+y) = cosxcosy - sinxsiny

Replace expansions into original fraction

`rArr (-2cosxsiny)/(cosxcosy -sinxsiny) xx(-1)/(-1)`

`= (2cosxsiny)/(sinxsiny -cosxcosy)`