Common Half angle identity:
1. #sin a = 2 sin (a/2)* cos (a/2)#
Half angle Identities in term of t = tan a/2.
2. #sin a = (2t)/(1 + t^2)#
3.#cos a = (1 - t^2)/(1 + t^2)#
- #tan a = (2t)/(1 - t^2).#
Use of half angle identities to solve trig equations.
Example. Solve #cos x + 2*sin x = 1 + tan (x/2).#
Solution. Call #t = tan (x/2)#. Use half angle identities (2) and (3) to transform the equation.
#(1 - t^2)/4 + (1 + t^2)/4 = 1 + t.#
#1 - t^2 + 4t = (1 + t)(1 + t^2)#
#t^3 + 2t^2 - 3t = t*(t^2 + 2t - 3) = 0.#
Next, solve the #3# basic trig equations: #tan (x/2) = t = 0; tan (x/2) = -3;# and #tan (x/2) = 1.#
