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

1 Answer
Jim G.
Apr 4, 2016

#(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) #

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