Question

Prove that cos(π/4-x)cos(π/4-y)-sin(π/4-x)sin(π/4-y)=sin(x+y)

Answer 1

Solution :

To Prove :

cos(π/4-x) cos(π/4-y) - sin(π/4-x) sin(π/4-y) = sin(x+y)

Proof :

LHS

= cos(π/4-x) cos(π/4-y) - sin(π/4-x) sin(π/4-y)

= cos ( π/4 - x + π/4 - y )

{ Using Formula ,
[ cos (A + B) = cosA cosB - sinA sinB ] }

= cos ( π/2 - x - y )

= cos [ π/2 - (x + y) ]

= sin (x + y) _ { ∵ cos(π/2 - θ) = sinθ }

= RHS

Hence Proved