 Hello and welcome to the session. In this session first we will discuss trigonometric functions of sum and difference of two angles. The basic results in this connection are called trigonometric identities. We have seen that sin of minus x is equal to minus sin x then cos of minus x is equal to cos x. Next we have cos of x plus y is equal to cos x multiplied by cos y minus sin x multiplied by sin y then cos of x minus y is equal to cos x multiplied by cos y plus sin x multiplied by sin y then we have cos of pi by 2 minus x is equal to sin x then sin of pi by 2 minus x is equal to cos x next we have sin of x plus y is equal to sin x multiplied by cos y plus cos x multiplied by sin y then sin of x minus y is equal to sin x multiplied by cos y minus cos x multiplied by sin y then next identity is cos of pi by 2 plus x is equal to minus sin x then sin of pi by 2 plus x is equal to cos x then next is cos of pi minus x is equal to minus cos x then sin of pi minus x is equal to sin x next is cos of pi plus x is equal to minus cos x then sin of pi plus x is equal to minus sin x then we have cos of 2 pi minus x is equal to cos x then sin of 2 pi minus x is equal to minus sin x next we have if none of the angles x y and x plus minus y is an odd multiple of pi by 2 then we have tan of x plus y is equal to tan x plus tan y upon 1 minus tan x into tan y and tan of x minus y is equal to tan x minus tan y upon 1 plus tan x into tan y and if none of the angles x y and x plus minus y is a multiple of pi then we have cot of x plus y is equal to cot x multiplied by cot y minus 1 upon cot y plus cot x then cot of x minus y is equal to cot x multiplied by cot y plus 1 upon cot y minus cot x next is cos 2 x is equal to cos square x minus sin square x equal to 2 cos square x minus 1 equal to 1 minus 2 sin square x equal to 1 minus tan square x upon 1 plus tan square x then sin 2x is equal to 2 sin x multiplied by cos x is equal to 2 tan x upon 1 plus tan square x next is tan 2x is equal to 2 tan x upon 1 minus tan square x then sin 3x is equal to 3 sin x minus 4 sin 2x cos 3x is equal to 4 cos 2x minus 3 cos x then tan 3x is equal to 3 tan x minus tan cube x upon 1 minus 3 tan square x next is cos x plus cos y is equal to 2 cos x plus y upon 2 multiplied by cos x minus y upon 2 next identity is cos x minus cos y is equal to minus 2 sin x plus y upon 2 multiplied by sin x minus y upon 2 then we have sin x plus sin y is equal to 2 sin x plus y upon 2 multiplied by cos x minus y upon 2 then the next identity is sin x minus sin y is equal to 2 cos x plus y upon 2 multiplied by sin x minus y upon 2 next is 2 cos x multiplied by cos y is equal to cos of x plus y plus cos of x minus y then minus 2 sin x sin y is equal to cos of x plus y minus cos of x minus y then 2 sin x cos y is equal to sin of x plus y plus sin of x minus y then 2 cos x sin y is equal to sin of x plus y minus sin of x minus y consider cos 5x plus cos 3x let's try and express this as product of sin of cos sin or sin n cos sin for this we can use the identity cos x plus cos y is equal to 2 cos x plus y upon 2 multiplied by cos x minus y upon 2 so here we have this would be equal to 2 cos x plus y upon 2 multiplied by cos x minus y upon 2 so this comes out to be equal to 2 cos 4x multiplied by cos x in this way we can use the different identities that we have just stated to evaluate a given trigonometric expression next we discuss trigonometric equations equations involving trigonometric functions of a variable are called trigonometric equations then solutions of a trigonometric equation for which we have x is greater than equal to 0 and less than 2 pi are called principal solutions the expression involving integer n which gives all solutions of a trigonometric equation is called the general solution let's try and find out the principal solution of the equation given by sin x equal to 1 upon 2 we know that sin pi by 6 is equal to 1 upon 2 and sin pi minus pi by 6 is equal to 1 upon 2 from here we have sin 5 pi by 6 is equal to 1 upon 2 and we already have sin pi by 6 is equal to 1 upon 2 thus the principal solutions are x equal to pi by 6 and 5 pi by 6 next we will try to find the general solution of trigonometric equations we already know that sin x equal to 0 gives x equal to n pi where n belongs to z then cos x equal to 0 gives x equal to 2n plus 1 pi by 2 where n belongs to z now we are following very important results it says for any real numbers x and y sin x equal to sin y implies that x is equal to n pi plus minus 1 to the power n y where this n belongs to z then for any real numbers x and y cos x equal to cos y implies that x is equal to 2n pi plus minus y where again n belongs to z also tan x equal to tan y implies x equal to n pi plus y where n belongs to z and x and y are not odd multiple of pi by 2 consider the trigonometric equation sin 2x equal to minus 1 upon 2 that is we have sin 2x is equal to minus sin pi by 6 or this could also be written as sin pi plus pi by 6 that is we now have sin 2x is equal to sin of 7 pi by 6 so this implies that 2x is equal to n pi plus minus 1 to the power n multiplied by 7 pi by 6 where n belongs to z that is this gives us that x is equal to n pi by 2 plus minus 1 to the power n 7 pi by 12 where again n belongs to z this is the general solution of the given trigonometric equation this completes the session hope you have understood the trigonometric identities and the trigonometric equations