2cos x sin y ... ( ) Let x + y = θ and x – y = φ . Therefore θ θ and +φ −φ Substituting the values of x and y in ( ), ( ), ( ) and ( ), we get cos θ + cos φ = cos θ θ cos +φ −φ cos θ – cos φ = – sin θ θ sin – + φ φ sin θ + sin φ = sin θ θ cos +φ −φ MATHEMATICS sin θ – sin φ = cos θ θ sin +φ −φ Since θ and φ can take any real values, we can replace θ by x and φ by y . Thus, we get cos x + cos y = cos cos ; cos x – cos y = – sin sin , sin x + sin y = sin cos ; sin x – sin y = cos sin Remark As a part of identities given in , we can prove the following results: .
cos x cos y = cos ( x + y ) + cos ( x – y ) – sin x sin y = cos ( x + y ) – cos ( x – y ) (iii) sin x cos y = sin ( x + y ) + sin ( x – y ) (iv) cos x sin y = sin ( x + y ) – sin ( x – y ). Example Prove that 3sin sec 4sin cot π π π π Solution We have L.H.S. = 3sin sec 4sin cot π π π π = × × – sin π π− × = – sin π = – × = = R.H.S. Example Find the value of sin °.