Prove trigonometric identities?

Oh man, I am so lost!

cos(x-y)/(cosx cosy) = 1+tanxtany

Any advice on where to start?

2 Answers
Jun 10, 2018

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

Jun 10, 2018

"see explanation"

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"