Trig Identities

Video lesson5 worked examples

Theory

Trigonometric identities are used to simplify expressions. While many exist, you are only required to memorise two for National 5:

\sin^2 x + \cos^2 x = 1

\tan x = \frac{\sin x}{\cos x}

Crucial Note: These formulas are not given on the exam formula sheet and must be memorised.

The Golden Rule: two moves solve almost everything — replace $\tan x$ with $\frac{\sin x}{\cos x}$ , and swap $\sin^2 x + \cos^2 x$ for $1$ (or the rearrangements $\sin^2 x = 1 - \cos^2 x$ and $\cos^2 x = 1 - \sin^2 x$ ). Always show the working — these questions carry a “show your working” instruction.

⚠️ Common Examiner Traps

Notation: $\sin^2 x$ means $(\sin x)^2$ , not $\sin(x^2)$ .
Same angle: the identity $\sin^2 x + \cos^2 x = 1$ only applies when both terms have the same angle.
Expanding a square: $(\sin x + \cos x)^2$ is a double bracket — it has a middle term $2\sin x\cos x$ , it is not $\sin^2 x + \cos^2 x$ .
Show working: a correct final answer with no steps can still lose marks here.

Worked examples

Example 1

Simplify $4\sin^2 x + 4\cos^2 x$ .

Step 1: Extract the common factor of 4:

4(\sin^2 x + \cos^2 x)

Step 2: Substitute the identity $\sin^2 x + \cos^2 x = 1$ :

4(1) = 4

Answer: 4.

Example 2

Show that $\cos x \tan x = \sin x$ .

Step 1: Substitute the identity for $\tan x$ :

\cos x \times \left(\frac{\sin x}{\cos x}\right)

Step 2: The $\cos x$ terms cancel out.

Answer: $\sin x$ .

Example 3

Simplify $\frac{1 - \cos^2 x}{\cos^2 x}$ .

Step 1: Substitute the rearranged identity $\sin^2 x = 1 - \cos^2 x$ :

\frac{\sin^2 x}{\cos^2 x}

Step 2: Substitute the identity $\tan x = \frac{\sin x}{\cos x}$ :

\left(\frac{\sin x}{\cos x}\right)^2 = \tan^2 x

Answer: $\tan^2 x$ .

Example 4

Expand and simplify $(\sin x + \cos x)^2$ . Show your working.

Step 1: Expand the double bracket, just like any binomial square:

\sin^2 x + 2\sin x\cos x + \cos^2 x

Step 2: Group the two squared terms and use $\sin^2 x + \cos^2 x = 1$ :

(\sin^2 x + \cos^2 x) + 2\sin x\cos x = 1 + 2\sin x\cos x

Answer: $1 + 2\sin x\cos x$ .

Example 5

🎯 Exam-style (express in a given form)

Express $3\cos^2 x - 1$ in the form $a + b\sin^2 x$ . Show your working.

Step 1: The target form uses $\sin^2 x$ , so replace $\cos^2 x$ using $\cos^2 x = 1 - \sin^2 x$ :

3(1 - \sin^2 x) - 1

Step 2: Expand and collect the constant terms:

3 - 3\sin^2 x - 1 = 2 - 3\sin^2 x

Answer: $2 - 3\sin^2 x$ , so $a = 2$ and $b = -3$ .

Ready to practise?

10 Trig Identities questions with answers and video solutions