Trigonometry Double angle formulas

In trigonometry, for any angle ‘a’, if we know the trigonometric ratios of ‘a’ we can find the corresponding ratios of the angle ‘2a’. The formulae that we use for finding that are called trig double angle formulas. What we mean by that is, that since we know the value of angle ‘a’, therefore we also know the values of sin(a), cos(a) and tan(a) (which can be easily found once we know the angle a). We can use these known values of sin(a), cos(a) and tan(a) to find the values of sin(2a), cos(2a) and tan(2a).

Double angle formulas trig for sin (2a):
Sin(2a) = 2 (sin a) (cos a)
Proof: Sin (2a) = sin (a+a)
= (sin a)(cos a) + (cos a)(sin a), that is using the angle sum identity which is like this:
Sin (a+b) = (sin a)(cos b) + (cos a)(sin b).
So now, sin (2a) = 2 (sin a) (cos a)
Hence the identity is proved.

Double angle formulas trig for cos (2a):
I] Cos (2a) = 2 cos^2 (a) – 1
Proof: Cos (2a) = cos (a+a)
= (cos a) (cos a) – (sin a) (sin a), that is using the angle sum identity for cos which is like this: cos(a+b) = (cos a) (cos b) – (sin a) (sin b).
Therefore now, cos (2a) = cos^2 (a) – sin^2 (a) -------------- (1)
We also know that cos^2(a) + sin^2(a) = 1,
Therefore sin^2(a) = 1 -  cos^2(a)
Substituting the above value of sin^2(a) into the equation number (1) we have,
Cos (2a) = cos^2 (a) – (1 – cos^2 (a))
= cos^2 (a) – 1 + cos^2 (a)
= 2 cos^2 (a) – 1
Hence identity proved. The same identity also has another form.
II) Cos (2a) = 1 – 2 sin^2 (a)

Proof: From the identity cos^2 (a) + sin^2 (a) = 1, we know that cos^2 (a) = 1 – sin^2 (a). Substituting this value of cos^2 (a) into the equation (1) we have:
Cos (2a) = 1 – sin^2 (a) – sin^2 (a) = 1 – 2 sin^2 (a)
Hence proved.

Double angle formula for tan(2a):
I) Tan (2a) = sin (2a) / cos (2a)
This one can be proved using double angle formulas for sin and cos from above.
II) Tan (2a) = 2 tan(a)/(1 – tan^2 (a))
Proof: Tan (2a) = tan (a + a) = (tan a + tan b)/(1 – (tan a)(tan b)) = (tan a + tan a)/(1 – (tan a)(tan a))
= 2 tan a/(1 – tan^2 (a))
Hence proved.