Prove trigonometric identities?
Oh man, I am so lost!
cos(x-y)/(cosx cosy) = 1+tanxtany
Any advice on where to start?
Oh man, I am so lost!
Any advice on where to start?
2 Answers
See explanation
Explanation:
We want to prove
cos(x-y)/(cos(x)cos(y))=1+tan(x)tan(y)
Remember the angle-difference identity
color(blue)((1) color(white)(BB)cos(x-y)=cos(x)cos(y)+sin(x)sin(y)
Thus
LHS=cos(x-y)/(cos(x)cos(y))
color(white)(LHS)=(cos(x)cos(y)+sin(x)sin(y))/(cos(x)cos(y)) larr "(1)"
color(white)(LHS)=(cos(x)cos(y))/(cos(x)cos(y))+(sin(x)sin(y))/(cos(x)cos(y))
color(white)(LHS)=1+((sin(x))/cos(x))(sin(y)/cos(y))
color(white)(LHS)=1+tan(x)tan(y)=RHS
Explanation:
"using the "color(blue)"trigonometric identity"
•color(white)(x)cos(x-y)=cosxcosy+sinxsiny
"consider the left side"
(cosxcosy+sinxsiny)/(cosxcosy)
=(cancel(cosxcosy))/cancel(cosxcosy)+(sinxsiny)/(cosxcosy)
=1+sinx/cosx xxsiny/cosy
=1+tanxtany=" right side"rArr"verified"