3.3 Trigonometry

cosec x=1/sin x, sec x=1/cos x and cot x=cos x/sin x. They are undefined where the denominator is zero and never equal zero themselves.

Use the original function’s sign and period to sketch reciprocal branches. Mark vertical asymptotes before plotting turning points.

sec x has asymptotes at π/2+nπ and reaches ±1 at x=nπ.

Reciprocal notation is not inverse-function notation: sec x is not cos⁻¹x.

Use fundamental, compound-angle and double-angle identities to replace a difficult expression with an equivalent one. The replacement is valid wherever the original and new expressions are defined.

Choose a target form first—for example, a single sine or a factorised square—and simplify one side only when proving an identity.

sin(π/2−x)=cos x converts a complementary-angle expression immediately.

A numerical agreement at a few angles is not proof, and cancellation can hide excluded denominator values.