To prove an identity, take one side and use true statements, line by line, until it becomes the other side. The question gives you the destination, so your job is the route. This appears often in the trigonometry section of Additional Mathematics, usually worded “show that” or “prove that”.
This lesson follows using a fundamental identity to rewrite an expression and sits inside trigonometric identities.
Why is starting from the answer unsafe?
Suppose you want to prove that −1 = 1, which is false. Square both sides and you get 1 = 1, which is true. A chain of correct-looking steps has turned a false statement into a true one, because squaring is not reversible.
The same can happen in a trig proof. If you start from “LHS = RHS” and work on both sides, you may finish with a true statement without proving the first one. Working on one side only removes the risk.
How do you build a proof?
- Pick the side with more going on. Fractions, mixed functions and brackets all signal the side to simplify.
- Convert to sin x and cos x when nothing more obvious works. tan, cot, sec and cosec all become fractions of sin x and cos x.
- Combine fractions over a common denominator. Keep the brackets.
- Look for an identity. Expressions like sin²x + cos²x or 1 − cos²x are the signal to swap.
- Factorise and cancel common factors. Cancel only factors of the whole numerator and denominator.
- Finish with the other side. Write each line so the next follows from it.
Worked example
Prove that (1 + cos x) / sin x + sin x / (1 + cos x) ≡ 2 / sin x.
Start with the left side. Use the common denominator sin x (1 + cos x):
[(1 + cos x)² + sin²x] / [sin x (1 + cos x)]
Expand the bracket: (1 + cos x)² = 1 + 2cos x + cos²x, so the numerator is 1 + 2cos x + cos²x + sin²x.
Use sin²x + cos²x = 1: the numerator becomes 1 + 2cos x + 1 = 2 + 2cos x = 2(1 + cos x).
Cancel the common factor (1 + cos x):
2(1 + cos x) / [sin x (1 + cos x)] = 2 / sin x
This is the right side, so the identity is proved. A check at x = 60°: the left side is 1.5 / 0.866 + 0.866 / 1.5 ≈ 1.732 + 0.577 = 2.309, and 2 / sin 60° ≈ 2.309.
The mistake to watch for
A common slip is to write the identity as an equation and manipulate both sides.
Mistaken working: Cross-multiply (1 + cos x)² + sin²x = 2(1 + cos x) and then simplify both sides until they match.
The student began by assuming the two sides are equal.
The correction is to choose the left side, transform it alone and reach the right side. The logic must run forwards: the first line is something you know, the last line is what you wanted. If you really do need to work both sides, each step must be reversible and you should say so, but a one-sided proof is the safer habit.
Check yourself
1. Prove that tan x + cot x ≡ sec x cosec x.
Show answer
LHS = sin x / cos x + cos x / sin x = (sin²x + cos²x) / (sin x cos x) = 1 / (sin x cos x) = (1 / cos x)(1 / sin x) = sec x cosec x.
2. Prove that sin⁴x − cos⁴x ≡ sin²x − cos²x.
Show answer
LHS = (sin²x − cos²x)(sin²x + cos²x) = (sin²x − cos²x) × 1 = sin²x − cos²x.
3. Prove that cos²x − sin²x ≡ 2cos²x − 1.
Show answer
Replace sin²x with 1 − cos²x: LHS = cos²x − (1 − cos²x) = cos²x − 1 + cos²x = 2cos²x − 1.
Where this leads next
Next, simplify an expression before substitution to see how the same identities make evaluation easier. The non-calculator working trainer and the quadratic structure explorer help with the algebra inside longer proofs.
If your working is correct but hard to organise, our teachers look at exactly that in online one-to-one Additional Mathematics tuition.