Use the Pythagorean identities: ( sin^2 x + cos^2 x = 1 ), ( 1 + tan^2 x = sec^2 x ), ( 1 + cot^2 x = csc^2 x )
Rewrite all trig functions in terms of sine and cosine
Factor out common terms from the numerator or denominator
Use reciprocal identities: ( sec x = frac{1}{cos x} ), ( csc x = frac{1}{sin x} ), ( cot x = frac{cos x}{sin x} ), ( tan x = frac{sin x}{cos x} )
Convert expressions to a common denominator
Apply algebraic factoring techniques such as grouping, difference of squares, or trinomials
Use conjugates to simplify expressions with radicals or sums/differences
Cancel common factors only after factoring completely
Replace ( sin^2 x ) or ( cos^2 x ) using Pythagorean identities when needed
Use angle sum and difference identities if the expression involves shifted angles
Verify the domain to avoid invalid cancellations or division by zero