Step by Step
1
Work only one side
Choose one side of the equation — usually whichever looks more complex — and transform ONLY that side, step by step, until it matches the other side exactly. Never move terms across the equals sign, which would treat the identity as something to solve rather than something to prove.
2
Convert everything to sine and cosine first
As a default strategy, rewrite every trig function in terms of sine and cosine — this often reveals simplifications or cancellations that aren't obvious when other functions (tan, sec, csc, cot) are still in the mix.
3
Apply Pythagorean and other known identities
Look for opportunities to substitute sin²θ+cos²θ=1 (or its rearranged forms), or other identities covered in earlier lessons, wherever they simplify the expression.
4
Combine fractions or use conjugates
If the expression has fractions, combine them over a common denominator. If an expression looks like 1−cos(θ) or 1+sin(θ) in a denominator, multiplying by the conjugate can help simplify using the Pythagorean identity.
Applied Walkthrough
1
Prove the identity: (1 − cos²θ)/sin(θ) = sin(θ). Work only the left side, since it's more complex.
2
Recognize that 1 − cos²θ is exactly the Pythagorean identity rearranged: 1 − cos²θ = sin²θ.
3
Substitute: (sin²θ)/sin(θ).
4
Simplify by canceling one factor of sin(θ): sin²θ/sin(θ) = sin(θ) — which now exactly matches the right side, completing the proof.
Exam Application
Exams test whether you follow the correct proof strategy (working only one side, never crossing the equals sign), and whether you can recognize when to apply Pythagorean identities or convert to sine/cosine to simplify the expression.
⚠ Common Trap
The most common trap is treating the identity like an equation to solve — moving terms across the equals sign or working on both sides simultaneously. A trig identity proof must transform only one side until it matches the other exactly.
✓ Quick Self-Check
1. What is the golden rule for proving a trig identity?
Work only one side of the equation — never move terms across the equals sign.
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2. What is a good default first strategy when proving a trig identity?
Convert every trig function to sine and cosine.
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3. Prove (1 − cos²θ)/sin(θ) = sin(θ) — what identity is used?
The Pythagorean identity, rearranged as 1 − cos²θ = sin²θ, then simplified by canceling a factor of sin(θ).
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4. What technique helps simplify a denominator like 1 − cos(θ)?
Multiplying by the conjugate, which often allows the Pythagorean identity to simplify the result.
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5. Why should you never cross the equals sign when proving an identity?
Because doing so treats the identity as something to solve rather than prove — the goal is to transform one side until it matches the other, not to manipulate both sides as if solving an equation.
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