Question

Prove trigonometric identities?

Oh man, I am so lost!

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

Any advice on where to start?

Answer 1

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`

Answer 2

`"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"`