~~~Proving Trigonometric Functions~~~

[yasaminG]

cos ( A+B) cos B+ sin (A+B) sin B=cos A

and

how do we know that Tan 2@=2tan@/1-tan^2@

i really appreciate your help :!:
thank you

 

[Gene]

I assume you have the sum of angle formulas
cos(a+b)=cos(a)cos(b)-sin(a)sin(b)
sin(a+b)=sin(a)cos(b)+cos(a)sin(b)
***************************

cos(a+b)cos(b)+sin(a+b)sin(b) =
cos(a)cos²(b)-sin(a)sin(b)cos(b) +
sin(a)sin(b)cos(b)+cos(a)sin²(b) =

cos(a){cos²(b)+sin²(b)} = cos(a)
***************************

tan(2@)=sin(@+@)/cos(@+@) =
2sin(@)cos(@)/(cos²(@)-sin²(@)) =
2sin(@)cos(@)/(cos²@)(1-sin²(@)/cos²(@)) =
2(sin(@)/cos(@))/(1-sin²(@)/cos²(@)) =
2tan(@)/(1-tan²(@))

Watch your ()s.
2tan(@)/1-tan²(@) is NOT the same as
2tan(@)/(1-tan²(@))
What you typed means
(2tan(@)/1)-tan²(@) =
2tan(@)-tan²(@)