A TRIGONOMETRIC IDENTITY is an equation involving the trigonometric functions that

holds for all values of the variable.

sin(a+b)=sin(a)cos(b)+cos(a)sin(b)

By taking L.H.S. :-sin (a+b) = sin [a-(-b)]

= sin(a) cos(-b) – cos(a) sin(-b) = sin(a) cos(b) – cos(a) [-sin(b)]

= sin(a) cos(b) + cos(a) sin(b) = R.H.S.

Hence Proved

sin(a-b) = sin(a)cos(b) - cos(a)sin(b)By taking L.H.S. :-

By taking L.H.S. :-

By taking R.H.S. :-

Multiply by 1 in the form of the conjugate of the denominator, so we get,

xxxx

sin1cossectan

xx

xx

sin1sin1

sin1cos

Now, by simplifying the numerator and the denominator, we get,

xxx

2sin1)sin1(cos

xxxx

2cossincoscos

‘Split’ the fraction and simplify

xxx

xx

22 cossincos

coscos

xx

x cossin

cos1

By using Reciprocal Identities, we get :-

xx tansec And then by using the commutative property of addition…

xx sectan = L.H.S. Hence Proved