Postby Ivanjam » Thu Sep 10, 2015 7:11 am
In general, you only need to know three trigonometric identities along with the definitions of tan(x), cot(x), sec(x), csc(x) through sin(x) and cos(x), as well as the fact that cos(x) is an even function: cos(-x)=cos(x), and that sin(x) is odd: sin(-x)=-sin(x). The basic identities are:
(1) The Pythagorean identity: (sin(x))^2+(cos(x))^2=1
(2) The angle sum formula for sine: sin(x+y)=sin(x)cos(y)+cos(x)sin(y)
(3) The angle sum formula for cosine: cos(x+y)=cos(x)cos(y)-sin(x)sin(y)
Now, let's play around with these basic identities:
(a) Divide identity (1) by (cos(x))^2 to obtain: (tan(x))^2+1=(sec(x))^2
(b) Divide identity (1) by (sin(x))^2 to obtain: 1+(cot(x))^2=(csc(x))^2
(c) In identity (2), let y=x. We obtain: sin(2x)=2sin(x)cos(x)
(d) In identity (3), let y=x. We obtain: cos(2x)=(cos(x))^2-(sin(x))^2
(e) In the RHS of identity (d), replace (sin(x))^2 by 1-(cos(x))^2 (from rearranging identity (1)). We obtain: cos(2x)=2(cos(x))^2-1
(f) Similarly, in identity (d), replace (cos(x))^2 by 1-(sin(x))^2 to obtain the alternative formula: cos(2x)=1-2(sin(x))^2
There are many more identities we can derive by playing around with identities (1)-(3), and (a)-(f).