cos x , cos y and cos ( x + y ) are non-zero. Now tan ( x + y ) = sin( cos( = sin cos cos sin cos cos sin sin Dividing numerator and denominator by cos x cos y , we have tan ( x + y ) = cos cos sin sin cos cos cos cos cos cos sin cos cos cos cos sin tan tan – tan tan . tan ( x – y ) = tan – tan + tan tan If we replace y by – y in Identity , we get tan ( x – y ) = tan [ x + (– y )] = tan tan( tan tan( = tan tan tan tan . If none of the angles x , y and ( x + y ) is a multiple of πππππ , then cot ( x + y ) = cot cot – cot +cot TRIGONOMETRIC FUNCTIONS Since, none of the x , y and ( x + y ) is multiple of π , we find that sin x sin y and sin ( x + y ) are non-zero.
Now, cot ( x + y )= cos ( cos cos – sin sin sin ( sin cos cos sin Dividing numerator and denominator by sin x sin y , we have cot ( x + y ) = cot cot – cot cot . cot ( x – y )= cot cot + cot – cot if none of angles x, y and x–y is a multiple of π If we replace y by – y in identity , we get the result . cos x = cos x – sin x = cos x – = – sin x = – tan + tan We know that cos ( x + y) = cos x cos y – sin x sin y Replacing y by x , we get cos x = cos x – sin