9 min readPathway sheetPart II (2 marks) and Part III (5 marks) proofs

Proving trigonometric identities

Start from the harder side, change everything to sin and cos if stuck, use the identities, and simplify until you reach the other side.

LHS and RHSThe left-hand side and the right-hand side of the equation

Golden ruleWork on one side only. Do not move terms across the = sign

Quotient ratiostan θ = sin θ/cos θ, cot θ = cos θ/sin θ

Conjugate trick(sec θ − tan θ)(sec θ + tan θ) = 1

A method that works for most proofs

1
Write the LHS and the RHS separately.
2
Start with the side that looks more complicated.
3★
If you are stuck, change tan, cot, sec and cosec into sin and cos.
4
Use an identity, or algebra such as a² − b² = (a + b)(a − b), or take an LCM.
5
Simplify one line at a time until you reach the other side.
6
End with '= RHS. Hence proved.'

★ key step

Algebra you will use

Rule	Example in trigonometry
a² − b² = (a + b)(a − b)	1 − sin²θ = (1 + sin θ)(1 − sin θ)
(a + b)² = a² + 2ab + b²	(sin θ + cos θ)² = 1 + 2 sin θ cos θ
(a − b)² = a² − 2ab + b²	(sin θ − cos θ)² = 1 − 2 sin θ cos θ
1/a + 1/b = (a + b)/(ab)	1/sin θ + 1/cos θ = (cos θ + sin θ)/(sin θ cos θ)

Example 1: change to sin and cos

Problem

Prove that sec θ cosec θ − cot θ = tan θ.

Working

LHS = sec θ cosec θ − cot θ
= 1/(cos θ sin θ) − cos θ/sin θ
= (1 − cos²θ)/(sin θ cos θ), taking the LCM sin θ cos θ
= sin²θ/(sin θ cos θ)
= sin θ/cos θ = tan θ = RHS

Answer

LHS = RHS. Hence proved.

Example 2: expand and add

Problem

Prove that (sin θ + cos θ)² + (sin θ − cos θ)² = 2.

Working

LHS = (sin²θ + 2 sin θ cos θ + cos²θ) + (sin²θ − 2 sin θ cos θ + cos²θ)
The two 2 sin θ cos θ terms cancel.
= 2 sin²θ + 2 cos²θ
= 2(sin²θ + cos²θ) = 2 × 1 = 2 = RHS

Answer

LHS = RHS. Hence proved.

Example 3: multiply by the conjugate

Problem

Prove that cos θ/(1 − sin θ) = sec θ + tan θ.

Working

Multiply the top and the bottom by (1 + sin θ). This is the conjugate of (1 − sin θ): the same two terms with the opposite sign between them.
LHS = cos θ(1 + sin θ)/((1 − sin θ)(1 + sin θ))
= cos θ(1 + sin θ)/(1 − sin²θ) = cos θ(1 + sin θ)/cos²θ
= (1 + sin θ)/cos θ
= 1/cos θ + sin θ/cos θ = sec θ + tan θ = RHS

Answer

LHS = RHS. Hence proved.

In real life

The software that rotates photos on your phone uses sin θ and cos θ. Because sin²θ + cos²θ = 1, the picture turns without stretching. Identities let engineers swap a long expression for a short one.

Exam angle

2 marks: “Prove that sin θ tan θ + cos θ = sec θ.” LHS = sin θ × (sin θ/cos θ) + cos θ = (sin²θ + cos²θ)/cos θ = 1/cos θ = **sec θ** = RHS.

5 marks: “Prove that (1 − sin θ)/(1 + sin θ) = (sec θ − tan θ)².” Start from the RHS: (1/cos θ − sin θ/cos θ)² = (1 − sin θ)²/cos²θ = (1 − sin θ)²/((1 − sin θ)(1 + sin θ)) = (1 − sin θ)/(1 + sin θ) = **LHS**.

5 marks: “Prove that cos θ/(1 − tan θ) + sin θ/(1 − cot θ) = sin θ + cos θ.” Change tan and cot into sin and cos: cos²θ/(cos θ − sin θ) − sin²θ/(cos θ − sin θ) = (cos²θ − sin²θ)/(cos θ − sin θ) = cos θ + sin θ = **RHS**.

Remember it

One side only; stuck? Go to sin and cos.