Construct proofs involving trigonometric functions and identities.

The idea

Constructing proofs involving trigonometric functions and identities requires manipulating expressions using known identities to show that one side of an equation is equivalent to the other. Start with the more complicated side and simplify using fundamental identities such as sin² θ + cos² θ = 1, tan θ = sin θ / cos θ, and the addition, double angle and harmonic form formulae. Techniques include factorising, combining fractions, multiplying numerator and denominator by a conjugate, and substituting equivalent expressions. For example, to prove that (1 − cos 2θ) / sin 2θ = tan θ, use cos 2θ = 1 − 2 sin² θ and sin 2θ = 2 sin θ cos θ, then simplify. Proofs must be logically sequenced, with each step justified by a named identity or algebraic operation, and must not assume the result being proved.

What earns the marks

• Use fundamental identities such as sin² θ + cos² θ = 1, tan θ = sin θ / cos θ, and the addition and double angle formulae to manipulate expressions.
• Start with one side of the identity (usually the more complex) and transform it step by step to match the other side, showing all working.
• Apply algebraic techniques such as factorising, combining fractions, and multiplying by conjugates to simplify trigonometric expressions.
• Justify each step by referencing the identity or algebraic rule used, and ensure the proof is coherent and logically complete.
• Avoid assuming the identity is true; work from one side to the other without cross-multiplying or moving terms across the equals sign.

Common mistakes and how to put them right

• Treating the identity as an equation and performing operations on both sides, such as cross-multiplying, which is not a valid proof method.

  Put it right: Manipulate one side only.

• Using incorrect identities, such as writing sin 2θ = 2 sin θ instead of 2 sin θ cos θ.

  Put it right: Double-check the identity being applied.

• Making algebraic errors when simplifying fractions, such as cancelling terms incorrectly.

  Put it right: Ensure you only cancel factors, not terms.

• Failing to show enough steps or justify each manipulation, leading to a gap in the proof.

  Put it right: Write clearly and reference identities used.

Exam tips

• Start with the more complicated side of the identity and simplify it to match the simpler side; this is usually easier than working from both sides.
• Write down the identities you plan to use before starting, and refer to them explicitly in your working.
• If you get stuck, try converting all trigonometric functions to sin and cos, then simplify.
• Check your proof by substituting a numerical value for θ (avoiding values where functions are undefined) to see if both sides are equal.