Identities

Welcome to the World of Identities!

Hi there! Welcome to one of the most useful chapters in your A Level Mathematics journey. In this section of Pure Mathematics: Trigonometry, we are going to look at Trigonometric Identities.

Think of an identity as a mathematical "alias." Just like a superhero might have a secret identity (like Peter Parker and Spider-Man), trigonometric expressions can often be written in a completely different way but still represent the exact same thing. Knowing these "aliases" allows you to simplify terrifyingly long equations into something much more manageable. Don't worry if this seems like a lot to memorize at first—with a few tricks and some practice, these will become second nature!

1. The Foundation: Year 1 Identities

Before we build the skyscraper, we need a solid ground. You should already be familiar with these two from your earlier studies. They are the "bread and butter" of trig identities.

The Tangent Identity

\( \tan \theta = \frac{\sin \theta}{\cos \theta} \)
Memory Aid: Think of Sin over Cos (Sun over Clouds). The sun is always above the clouds!

The Pythagorean Identity

\( \sin^2 \theta + \cos^2 \theta = 1 \)
This is just Pythagoras' Theorem hiding in a circle! In a unit circle where the hypotenuse is 1, the opposite side is \( \sin \theta \) and the adjacent side is \( \cos \theta \). So, \( a^2 + b^2 = c^2 \) becomes our identity.

Common Mistake: Watch out for the notation. Remember that \( \sin^2 \theta \) means \( (\sin \theta)^2 \). It is not the same as \( \sin(\theta^2) \)!

Key Takeaway: These two identities work for any value of \( \theta \). You can use them to swap back and forth between different trig functions to make equations easier to solve.

2. The New Reciprocals: Sec, Cosec, and Cot

In A Level, we meet three new functions. These are just the "upside-down" versions (reciprocals) of the ones you already know.

1. Cosecant: \( \text{cosec } \theta = \frac{1}{\sin \theta} \)
2. Secant: \( \sec \theta = \frac{1}{\cos \theta} \)
3. Cotangent: \( \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \)

Quick Trick to Remember: Look at the third letter of the new function name!
- cosec goes with sin
- sec goes with cos
- cot goes with tan

New Identities from Old Ones

If you take the basic identity \( \sin^2 \theta + \cos^2 \theta = 1 \) and divide everything by \( \cos^2 \theta \), you get:
\( \tan^2 \theta + 1 = \sec^2 \theta \)

If you divide the basic identity by \( \sin^2 \theta \), you get:
\( 1 + \cot^2 \theta = \text{cosec}^2 \theta \)

Did you know? You don't actually need to memorize these last two if you know how to derive them by dividing. If you blank in an exam, just write down \( \sin^2 \theta + \cos^2 \theta = 1 \) and start dividing!

Key Takeaway: Sec, Cosec, and Cot follow the same rules as Sin, Cos, and Tan, but they are undefined whenever their "bottom" part is zero.

3. Compound Angle Formulae

Sometimes we have two different angles (let's call them \( A \) and \( B \)) added together inside a trig function. We can't just "multiply out" the brackets like normal algebra. Instead, we use these special recipes:

Sine Compound:
\( \sin(A \pm B) = \sin A \cos B \pm \cos A \sin B \)
(Notice: Sine is "friendly"—it keeps the same sign \( + \) stays \( + \), and it mixes the Sin and Cos together.)

Cosine Compound:
\( \cos(A \pm B) = \cos A \cos B \mp \sin A \sin B \)
(Notice: Cosine is "selfish"—it keeps the Cosines together and the Sines together, and it changes the sign! \( + \) becomes \( - \).)

Tangent Compound:
\( \tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B} \)

Encouraging Phrase: These look a bit like alphabet soup at first, but you'll usually be given these in a formula booklet. The key is knowing when to use them—usually when you see two different angles inside one function!

Key Takeaway: Compound angle formulae allow us to break down complex arguments like \( (45^\circ + 30^\circ) \) into exact values we already know.

4. Double Angle Formulae

What happens if the two angles are the same? If we replace \( B \) with \( A \) in the compound angle formulae, we get the Double Angle Formulae. These are incredibly common in exam questions!

1. Double Sine:
\( \sin 2\theta = 2 \sin \theta \cos \theta \)

2. Double Cosine (The Triple Threat):
This one is special because there are three ways to write it. They are all correct!
- \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \)
- \( \cos 2\theta = 2 \cos^2 \theta - 1 \)
- \( \cos 2\theta = 1 - 2 \sin^2 \theta \)

3. Double Tangent:
\( \tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta} \)

Quick Review Box:
If an equation has a mix of \( \cos 2\theta \) and \( \sin^2 \theta \), use the version of the identity that only has \( \sin \): \( 1 - 2 \sin^2 \theta \). This lets you create a quadratic equation that you can actually solve!

Key Takeaway: Use double angle identities to change an expression with \( 2\theta \) into one with just \( \theta \). This "harmonizes" the angles so you can solve the equation.

5. The Harmonic Form: \( R \sin(\theta \pm \alpha) \)

Have you ever seen an expression like \( 3 \sin \theta + 4 \cos \theta \) and thought, "I wish that was just one single sine wave"? Well, it can be!

We can write \( a \sin \theta \pm b \cos \theta \) in the form \( R \sin(\theta \pm \alpha) \).
Or we can write \( a \cos \theta \pm b \sin \theta \) in the form \( R \cos(\theta \mp \alpha) \).

Step-by-Step Process:

Step 1: Find \( R \). This is just Pythagoras! \( R = \sqrt{a^2 + b^2} \).
Step 2: Find \( \alpha \). Use the tangent: \( \tan \alpha = \frac{b}{a} \). (Be careful to match the coefficients correctly based on the expansion).
Step 3: Write it out. Use your new \( R \) and \( \alpha \) to rewrite the expression.

Why do we do this?
- It makes solving equations like \( 3 \sin \theta + 4 \cos \theta = 2 \) much easier.
- It lets you find the maximum and minimum values instantly. The maximum of \( R \sin(\theta + \alpha) \) is just \( R \), and the minimum is \( -R \).

Key Takeaway: The \( R \)-form turns two different waves into one wave. It’s like taking a messy chord and finding the single note it represents.

6. Strategies for Proving Identities

You will often be asked to "Prove that the LHS = RHS." This can be intimidating, but here is a simple battle plan:

1. Start with the "messier" side. It is much easier to simplify a complicated expression than it is to grow a simple one.
2. Change everything to Sine and Cosine. If you see Sec, Cosec, or Cot, swap them for their 1/Sin or 1/Cos equivalents. Often, things will start to cancel out!
3. Look for "Square" terms. If you see a \( \sin^2 \theta \) or a \( 1 \), think about the Pythagorean identity.
4. Match the angles. If one side has \( 2\theta \) and the other has \( \theta \), use a double angle formula immediately.
5. Common Denominators. If you have two fractions, add them together into one. The numerator often turns into a helpful identity.

Encouraging Phrase: Don't worry if your first attempt at a proof doesn't work. Sometimes you take a path that leads nowhere—just go back to the start and try a different "alias"! Every mistake teaches you which identity not to use next time.

Key Takeaway: Proofs are about logical steps. Show every step clearly, and always state which identity you are using if you want to impress the examiner!

Final Summary

- Identities are just different ways of writing the same thing.
- Sec, Cosec, and Cot are the reciprocals of Cos, Sin, and Tan.
- Compound and Double Angles help us deal with complicated arguments inside the trig functions.
- The R-form is your best friend for finding maximum values and solving "mixed" sine and cosine equations.

Keep practicing, and soon you'll be spotting these patterns everywhere. You've got this!