Worked example

Verify tan A + tan B = sin(A + B)/(cos A cos B)

Answer

Common denominator, then recognise the sine sum.

tan A + tan B
Left side.
= sin A/cos A + sin B/cos B
Quotient identity on both terms. · Quotient identity (tangent)
= (sin A cos B + cos A sin B)/(cos A cos B)
Common denominator.
= sin(A + B)/(cos A cos B)
Numerator is the sine sum formula. · Sine of a sum

tan A + tan B = sin(A + B) / (cos A cos B)

sin(A + B) = sin A cos B + cos A sin B

tan θ = sin θ / cos θ

Tangent sum as a single fraction — When a sum of two tangents has to become one fraction — standard in verification exercises, because the right side collapses two terms into one.
Sine of a sum — The angle itself is a sum and you need the functions separately — to evaluate exactly (sin 75° = sin(45° + 30°)), to expand, or to recognise the pattern running backwards.
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.

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.