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"#