Worked example

Verify tan θ + cot θ = sec θ csc θ

Answer

Convert to sin and cos, take the common denominator, and the Pythagorean identity closes it.

tan θ + cot θ
Left side — two different function families, so convert first.
= sin θ/cos θ + cos θ/sin θ
Quotient identities. · Quotient identity (tangent), Quotient identity (cotangent)
= (sin²θ + cos²θ)/(sin θ cos θ)
Common denominator, cross-multiply.
= 1/(sin θ cos θ)
Pythagorean identity in the numerator. · Pythagorean identity
= csc θ · sec θ
Split the reciprocal — matches the right side. · Reciprocal identity (cosecant), Reciprocal identity (secant)

This is the single most-assigned verification in a precalculus course, and it uses no formula beyond definitions plus sin² + cos² = 1.

tan θ = sin θ / cos θ

cot θ = cos θ / sin θ

sec θ = 1/cos θ

csc θ = 1/sin θ

sin²θ + cos²θ = 1

Quotient identity (tangent) — The single most common first step in a verification: replace tan θ with sin θ/cos θ so the whole expression is in one function family and fractions can be combined.
Quotient identity (cotangent) — Same idea as tan = sin/cos with the roles swapped: use it when cot θ and csc θ appear together, because both then share the denominator sin θ.
Reciprocal identity (secant) — Convert sec θ to 1/cos θ whenever it shares an expression with tan θ, since tan θ is already sin/cos and a common denominator then appears.
Reciprocal identity (cosecant) — Any time csc θ is in your way, the first move is to replace it with 1/sin θ. The reciprocal functions exist for notation, not for algebra — the algebra happens in sin and cos.
Pythagorean identity — Two squared trig terms of the same angle are added: the pair is exactly 1, so replace both with 1 — or, going the other way, replace a stubborn 1 with sin²θ + cos²θ to get a common denominator.

Open the verifier with the main identity pre-filled — substituting your own numbers into a step is the fastest way to find out which line you did not follow.