DWG NO. 5.4 — Lesson 4 of 5

Proving Trigonometric Identities

Unit 5: Trigonometric Identities and Equations · ~20 min

Lessons 5.1–5.3 handed you identities to use. This lesson is about building them yourself — taking a claimed identity and showing it's actually true for every angle. The biggest mindset shift: an identity proof is not an equation to solve. You never add, subtract, multiply, or divide both sides by the same thing, because you don't yet know the two sides are equal — that's the very thing you're trying to establish. Instead, you pick one side (usually the more complicated one) and transform it, step by legal step, until it becomes identical to the other side.

Strategy toolkit

• Work on one side at a time — almost always the more complicated side, leaving the simpler side as the fixed target.
• Convert everything to sine and cosine — secant, cosecant, tangent, and cotangent all have sin/cos definitions, and mixed expressions usually simplify once every term speaks the same language.
• Watch for Pythagorean substitutions — seeing 1 − sin²x or sec²x − 1 in an expression is a signal to swap it for cos²x or tan²x.
• Combine fractions over a common denominator before trying to simplify a sum or difference of trig fractions.
• Multiply by a conjugate when you see a sum like 1 + sinx in a denominator — multiplying by \(\frac{1 - sinx}{1 - sinx}\) often triggers a Pythagorean identity.

Guided practice

• 1.Prove sinx cotx = cosx.
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  sinx · (\(\frac{cosx}{sinx}\)) = cosx. The sinx factors cancel directly.
• 2.Show how the identity 1 + tan²x = sec²x can be derived directly from sin²x + cos²x = 1.
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  Divide every term of sin²x + cos²x = 1 by cos²x: \(\frac{sin^{2}x}{cos^{2}x}\) + 1 = \(\frac{1}{cos^{2}x,}\) which is tan²x + 1 = sec²x.
• 3.Prove cscx − sinx = cosx cotx.
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  \(\frac{1}{sinx}\) − sinx = \(\frac{1-sin^{2}x}{sinx}\) = \(\frac{cos^{2}x}{sinx}\) = cosx·(\(\frac{cosx}{sinx}\)) = cosx cotx. Confirmed.
• 4.Prove \(\frac{cosx}{1 - sinx}\) = secx + tanx.
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  Multiply top and bottom by the conjugate (1+sinx): cosx\(\frac{1+sinx}{1-sin^{2}x}\) = cosx\(\frac{1+sinx}{cos^{2}x}\) = \(\frac{1+sinx}{cosx}\) = \(\frac{1}{cosx}\) + \(\frac{sinx}{cosx}\) = secx + tanx. Confirmed.
• 5.Explain why squaring both sides is avoided when proving an identity.
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  Squaring can make two expressions look equal even when they weren't equal to begin with (for instance, x = −x is false, but x² = (−x)² is true). A step that can turn a false statement true is not a valid identity-proof move, since it would let you "prove" things that aren't actually identities.