Using Identities to Solve Equations

A trig equation with both sin(x) and cos(x) in it resists ordinary algebra, because you have two different functions of x tangled together. An identity gives you a legal substitution that is always true, so you can replace one function with an expression in the other without changing the equation's meaning. Once everything is written in terms of a single function, the equation becomes a polynomial in that function, and you solve it the way you'd solve any polynomial: factor, set each factor to zero, check the domain.

The most common tool here is the Pythagorean identity, sin²(x) + cos²(x) = 1, which lets you swap sin²(x) for 1 - cos²(x) or the reverse. Take 2 sin²(x) + cos(x) = 1 on the interval 0 to 2pi. Replace sin²(x) with 1 - cos²(x): 2(1 - cos²(x)) + cos(x) = 1, which simplifies to -2cos²(x) + cos(x) + 1 = 0. Let u = cos(x), factor the quadratic, solve for u, then solve cos(x) = u for x on the given interval. Every step after the substitution is algebra you already trust.

This matters because it connects identities, which are always true, to equations, which are only true at specific angles. An identity like sin²(x) + cos²(x) = 1 holds for every real x, so using it never loses or adds solutions. The equation you're solving, by contrast, picks out only the x-values that make a particular statement true. Treating an identity as a universal rewriting rule, while still checking the final answers against the original equation's domain, is what lets this method sit safely between the algebra of earlier units and the more complex trig equations ahead.