E: Trigonometry: Construct proofs involving trigonometric functions and identities.

The idea

This topic requires you to build logical arguments that establish trigonometric identities or solve equations by proof. You must use fundamental identities such as sin²θ + cos²θ ≡ 1, tanθ ≡ sinθ/cosθ, and the addition formulae, along with algebraic manipulation. A typical proof might ask you to show that (1 - cos²θ)/sinθ ≡ sinθ for all θ where defined. You would start with the left-hand side, replace 1 - cos²θ with sin²θ, then simplify to sinθ. Proofs must be clear, with each step justified, and you must avoid assuming the result. You also need to handle domains and restrictions, such as cosθ ≠ 0 when using tanθ. The skill is assessed through written responses where logical flow and correct use of identities are key.

What earns the marks

• Correctly apply the Pythagorean identity sin²θ + cos²θ ≡ 1 in either direction to simplify expressions.
• Use the quotient identity tanθ ≡ sinθ/cosθ appropriately, ensuring cosθ ≠ 0 where required.
• Manipulate expressions algebraically, such as factorising, combining fractions, or multiplying by conjugates, to transform one side into the other.
• Use addition and double-angle formulae correctly to expand or simplify trigonometric expressions.
• Maintain logical flow by working from one side of the identity to the other, or by transforming both sides to a common form, without assuming the equality.
• State any necessary domain restrictions or conditions for the identity to hold.

Common mistakes and how to put them right

• Assuming the identity is true and manipulating both sides simultaneously without justification.

  Put it right: Work from one side (usually the more complex) and transform it step-by-step to match the other side.

• Incorrectly applying identities, such as writing sin²θ + cos²θ ≡ 2 or misusing double-angle formulae.

  Put it right: Memorise and verify identities using known values, and practise applying them in simple cases.

• Ignoring domain restrictions, e.g., cancelling cosθ without noting cosθ ≠ 0.

  Put it right: Consider where expressions are defined and state restrictions explicitly in proofs.

Exam tips

• Start with the more complicated side of the identity and simplify it to match the other side; this often makes the proof clearer.
• Write each step on a new line and provide a brief reason (e.g., 'using sin²θ + cos²θ ≡ 1') to show your reasoning.
• If stuck, convert all trigonometric functions to sin and cos, then simplify algebraically; this can reveal a path forward.