sin y . cos ( π – ) = sin x If we replace x by π and y by x in Identity ( ), we get cos ( π ) = cos π cos x + sin π sin x = sin x . . sin ( π – ) = cos x Using the Identity , we have sin ( π ) = cos π π = cos x .
. sin ( x + y ) = sin x cos y + cos x sin y We know that sin ( x + y ) = cos π = cos π = cos ( π ) cos y + sin π sin y = sin x cos y + cos x sin y . sin ( x – y ) = sin x cos y – cos x sin y If we replace y by – y , in the Identity , we get the result. .
By taking suitable values of x and y in the identities , , and , we get the following results: cos π = – sin x sin π = cos x cos ( πππππ – x ) = – cos x sin ( πππππ – x ) = sin x MATHEMATICS cos ( πππππ + x ) = – cos x sin ( πππππ + x ) = – sin x cos ( πππππ – x ) = cos x sin ( πππππ – x ) = – sin x Similar results for tan x , cot x , sec x and cosec x can be obtained from the results of sin x and cos x . . If none of the angles x , y and ( x + y ) is an odd multiple of π , then tan ( x + y ) = tan + tan – tan tan Since none of the x , y and ( x + y ) is an odd multiple of π , it follows that