Proof involving trigonometric functions

Welcome to the World of Trigonometric Proofs!

In this chapter, we are going to learn how to prove that two trigonometric expressions are exactly the same, no matter what value you plug in for the angle. Think of it like being a mathematical detective: you start with one expression and use "clues" (identities) to show it is actually identical to another expression. This is a vital skill for OCR A Level Mathematics A (H240) because it builds the logical thinking you'll need for calculus and beyond.

Don't worry if this seems tricky at first! Proofs can feel a bit like puzzles. Sometimes you’ll take a path that leads nowhere, and that’s okay. The more you practice, the more you'll start to "see" the patterns.

Section 1: Your Trigonometric Toolbox

Before we start proving, we need our tools. These are the identities you are expected to know from your Stage 1 and Stage 2 studies. A trigonometric identity is an equation that is true for all values of the angle.

The "Essential Eight" Identities

1. The Quotient Identity: \(\tan \theta \equiv \frac{\sin \theta}{\cos \theta}\)
2. The Pythagorean Identity: \(\sin^2 \theta + \cos^2 \theta \equiv 1\)
3. Secant: \(\sec \theta \equiv \frac{1}{\cos \theta}\)
4. Cosecant: \(\text{cosec } \theta \equiv \frac{1}{\sin \theta}\)
5. Cotangent: \(\cot \theta \equiv \frac{1}{\tan \theta} \equiv \frac{\cos \theta}{\sin \theta}\)
6. The "Sec" Squared Identity: \(1 + \tan^2 \theta \equiv \sec^2 \theta\)
7. The "Cosec" Squared Identity: \(1 + \cot^2 \theta \equiv \text{cosec}^2 \theta\)
8. Double Angle Formulae:
- \(\sin 2\theta \equiv 2\sin \theta \cos \theta\)
- \(\cos 2\theta \equiv \cos^2 \theta - \sin^2 \theta \equiv 2\cos^2 \theta - 1 \equiv 1 - 2\sin^2 \theta\)

Quick Review: Remember the symbol \(\equiv\) means "Identically equal to." It’s stronger than a regular equals sign!

Memory Aid: For the squared identities, remember: "A Tan gent goes with a Sec ond" (\(1 + \tan^2 = \sec^2\)) and "A Cot goes with a Cosec" (\(1 + \cot^2 = \text{cosec}^2\)).

Key Takeaway: You cannot do proofs if you don't know these by heart. Spend 10 minutes every day writing them out until they are second nature!

Section 2: The Rules of the Game

When you are asked to "Prove" or "Show that," there are a few strict rules you must follow to get full marks from OCR examiners.

The "Start on One Side" Rule

The biggest mistake students make is treating a proof like an equation. Do not move terms from one side to the other. Instead:

1. Pick the Left-Hand Side (LHS) or the Right-Hand Side (RHS). It’s usually easier to start with the side that looks more complicated.
2. Manipulate that side using your identities.
3. Keep going until it looks exactly like the other side.
4. End with a concluding statement like "LHS = RHS, therefore the identity is proven."

Logical Steps

A proof is a series of logical steps. Each line must follow clearly from the one before it. If you skip too many steps, the examiner might think you've just "guessed" the answer.

Did you know? In Greek, Trigonometry means "Triangle Measure." Even though we use these fancy formulas, they all originally came from measuring the sides of right-angled triangles!

Section 3: Strategies for Success

If you're stuck on a proof, try these "battle-tested" strategies:

1. The "Sins and Cosses" Strategy: If you see \(\tan, \sec, \text{cosec, or } \cot\), change them all into expressions involving \(\sin\) and \(\cos\). This often reveals how things can be cancelled out.
2. Common Denominators: If you have two fractions, add them together by finding a common denominator. This is a classic OCR exam trick!
3. Look for Squares: If you see a \(\sin^2 \theta\) or a \(\cos^2 \theta\), think about the identity \(\sin^2 \theta + \cos^2 \theta \equiv 1\). You can rearrange this to \(\sin^2 \theta \equiv 1 - \cos^2 \theta\).
4. Factorise: Sometimes you can pull out a common factor to simplify the expression.

Common Mistake to Avoid: Writing \(\sin\) without an angle. \(\sin\) by itself is just a word; \(\sin \theta\) is a number. Always include the \(\theta\) or \(x\)!

Key Takeaway: If you are lost, turn everything into Sine and Cosine. It works about 80% of the time!

Section 4: Step-by-Step Worked Example

Let's look at an example similar to the one in the OCR syllabus:

Example: Prove that \(\frac{1}{\tan \theta + \cot \theta} \equiv \sin \theta \cos \theta\)

Step 1: Pick a side. The LHS is much messier, so let's start there.
LHS: \(\frac{1}{\tan \theta + \cot \theta}\)

Step 2: Use the "Sins and Cosses" strategy. Replace \(\tan\) and \(\cot\).
\(\frac{1}{\frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta}}\)

Step 3: Combine the fractions in the denominator. Find the common denominator, which is \(\sin \theta \cos \theta\).
\(\frac{1}{\frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta}}\)

Step 4: Use the Pythagorean identity. We know that \(\sin^2 \theta + \cos^2 \theta = 1\).
\(\frac{1}{\frac{1}{\sin \theta \cos \theta}}\)

Step 5: Simplify the fraction. Dividing 1 by a fraction is the same as flipping the fraction.
\(\sin \theta \cos \theta\)

Step 6: Conclusion.
LHS = RHS. Proven.

Quick Review: Notice how we didn't touch the RHS at all? We just worked on the LHS until it transformed into the RHS.

Section 5: Compound and Double Angle Proofs

As you progress to Stage 2, you will be asked to prove identities using Addition Formulae like \(\sin(A+B)\). The syllabus example asks to prove something using \(\cos^2(\theta + 45^\circ)\).

To do this, you first use the addition formula to expand the bracket:
\(\cos(A+B) = \cos A \cos B - \sin A \sin B\)

Then, you use the exact values you learned (like \(\sin 45^\circ = \frac{1}{\sqrt{2}}\)) to simplify the numbers. Finally, you might use a Double Angle Formula to reach the finish line.

Encouraging Phrase: These longer proofs are just several small steps put together. Tackle one bracket at a time, and the "big picture" will start to emerge!

Chapter Summary - Your Final Checklist

1. Memorise your core identities (Pythagoras, Quotients, Reciprocals, and Double Angles).
2. Start on one side only (the "messy" side).
3. Show every step clearly—don't skip the algebra!
4. Substitute \(\sin\) and \(\cos\) if you get stuck.
5. Conclude by stating that your working has reached the other side.

Key Takeaway: Trigonometric proof is about patience and substitution. If one identity doesn't work, try another!