How to Solve Identity Equations?

Identity function is a function which gives similar Functions or similar parameters which were used in the argument. It is also called as identity map or identity transformation. When identity function is applied to n – dimensional vector spaces then it is called as linear operator.
Mathematically, we can write identity function as F (y) = y.
Let 'C' be a collection or a Set then identity function 'f' on 'C' can be defined as;
F (y) = y for all elements of 'y' in 'C' that is the function is assigned to each element y of C.
Identity function F on C is normally denoted by idC.
We are going to solve identity equation using Trigonometric Identities in which we have to prove whether given equation is true or false.
Following example can be used for understanding how to solve identity equations with the help of trigonometric identities.
Consider the following identity equation:
sin² (2θ) - sin² (θ) = 0,
Following trigonometric identity will be used for solving above equation:
sin (2θ) = 2 sin(θ) cos (θ)
Rewriting the given equation as (sin (2θ))² - sin² (θ) = 0
2 sin (θ) cos (θ)² - sin² (θ) = 0,
4 sin² (θ) cos²(θ) - sin²(θ) = 0 (on expansion of the bracket),
sin² (θ) (4 cos² (θ) - 1) = 0 (taking out the common term ),
here the two solutions may be equal to zero.
1) sin² (θ) = 0,
sin (θ) = 0,
which gives θ= 0, π, 2π, ...
 
2) 4 cos² (θ) - 1 = 0,
cos² (θ) = ¼,
cos (θ) = ± ½,
θ= π/3, 2π/3, 4π/3, 5π/3, ...
 
Hence solutions will be as follows:
θ= 0 + πk,
θ= π/3 + πk,
θ= 2π/3 + πk.