Worked example

Verify (1 + tan²θ)cos θ = sec θ

Steps

1. (1 + tan²θ)cos θ
   Left side.
2. = sec²θ · cos θ
   Pythagorean identity in the tangent form. · Pythagorean identity (tangent form)
3. = (1/cos²θ) · cos θ
   Reciprocal identity — the square matters. · Reciprocal identity (secant)
4. = 1/cos θ
   One cosine cancels.
5. = sec θ
   Which is the right side, so the identity holds (θ ≠ 90° + k·180°).

The stumble in step 4 is the real student experience; the corrected line is kept visible because that is where the mistake happens.

Identities used

• Pythagorean identity (tangent form) — You see 1 + tan²θ — most often under a square root, √(1 + tan²θ), or as the product sec²θ appearing in a derivative. Replace the whole sum with sec²θ, or a lone sec²θ with 1 + tan²θ when the tangent form is what you need.
• 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.

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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