9 min readPathway sheet2 or 3-mark proof; sometimes part of a 5-mark question

Proving simple identities

To prove an identity, start from the harder side and simplify it until it equals the other side. Use the three identities, starting with sin²A + cos²A = 1.

Golden ruleWork on one side only; never move terms across the = sign

First move when stuckChange every ratio into sin and cos

Common toolsa² − b² = (a − b)(a + b), taking the LCM, the three identities

How to prove an identity

1
Write LHS and RHS separately.
2
Choose the more complicated side, usually the one with fractions or sums.
3★
Change to sin and cos, or use an identity directly.
4
Simplify: take the LCM, factorise, cancel common factors.
5
Reach the other side and write = RHS. Hence proved.

★ key step

Moves that unlock most proofs

When you see	Try this
1 − sin²A or 1 − cos²A	Replace it with cos²A or sin²A
sec²A − 1 or cosec²A − 1	Replace it with tan²A or cot²A
Two fractions added	Take the LCM
1 − sin A or 1 − cos A in a denominator	Multiply top and bottom by 1 + sin A or 1 + cos A
tan, cot, sec, cosec mixed together	Write everything in sin and cos
A difference of two squares	Factorise: a² − b² = (a − b)(a + b)

Proof by changing to sin and cos

Problem

Prove that (cosec A − cot A)(1 + cos A) = sin A.

Working

LHS = (cosec A − cot A)(1 + cos A)
= (1/sin A − cos A/sin A)(1 + cos A)
= ((1 − cos A)/sin A) × (1 + cos A)
= (1 − cos²A)/sin A
= sin²A/sin A (since 1 − cos²A = sin²A)
= sin A = RHS. Hence proved.

Answer

The LHS reduces to sin A, which is the RHS.

Proof using the conjugate

Problem

Prove that sin θ/(1 − cos θ) = cosec θ + cot θ.

Working

LHS = sin θ/(1 − cos θ)
Multiply top and bottom by (1 + cos θ): = sin θ (1 + cos θ)/((1 − cos θ)(1 + cos θ))
= sin θ (1 + cos θ)/(1 − cos²θ) = sin θ (1 + cos θ)/sin²θ
= (1 + cos θ)/sin θ
= 1/sin θ + cos θ/sin θ = cosec θ + cot θ = RHS. Hence proved.

Answer

Both sides are equal, so the identity is proved.

Mistakes that lose marks

Writing sin A + cos A = 1 instead of sin²A + cos²A = 1.
Cancelling a term joined by + or −, for example cancelling cos A in (1 + cos A)/cos A.
Working on both sides at once and moving terms across the = sign.
Forgetting to write LHS, RHS and Hence proved.

In real life

Checking a kirana shop bill two ways, item by item or in groups, should give the same total. Proving an identity is showing that two forms always give the same value.

Exam angle

1 marks: “Simplify (1 + tan²θ)(1 − sin θ)(1 + sin θ).” = sec²θ × (1 − sin²θ) = sec²θ × cos²θ = 1.

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

3 marks: “Prove that sec⁴A − sec²A = tan⁴A + tan²A.” LHS = sec²A (sec²A − 1) = (1 + tan²A) tan²A = tan²A + tan⁴A = RHS.

Remember it

LHS to RHS, one side at a time. When stuck, turn everything into sin and cos.