Worked example

Verify cot θ + csc θ = (1 + cos θ)/sin θ

Steps

1. cot θ + csc θ
   Left side.
2. = cos θ/sin θ + 1/sin θ
   Quotient and reciprocal identities. · Quotient identity (cotangent), Reciprocal identity (cosecant)
3. = (cos θ + 1)/sin θ
   Common denominator.
4. = (1 + cos θ)/sin θ
   Reordered — matches the right side.

Identities used

• 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 (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.

Check it yourself

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.

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