Proving a Trigonometric Identity

Identity proofs are the trig question students most fear, but they follow one reliable discipline: start from the more complicated side and work down to the other, never both at once. The engine is almost always $\sin^2\theta + \cos^2\theta = 1$.

Prove the identity $\dfrac{\sin\theta}{1 + \cos\theta} + \dfrac{1 + \cos\theta}{\sin\theta} \equiv \dfrac{2}{\sin\theta}$. [4]

The working

Start from the left-hand side (it is the busier one) and combine over a common denominator $\sin\theta(1 + \cos\theta)$: $\text{LHS} = \frac{\sin\theta}{1 + \cos\theta} + \frac{1 + \cos\theta}{\sin\theta} = \frac{\sin^2\theta + (1 + \cos\theta)^2}{\sin\theta(1 + \cos\theta)} \quad \text{(M1)}$

Expand the numerator: $\sin^2\theta + (1 + \cos\theta)^2 = \sin^2\theta + 1 + 2\cos\theta + \cos^2\theta \quad \text{(M1)}$

Now use $\sin^2\theta + \cos^2\theta = 1$ on the two squared terms: $= \underbrace{(\sin^2\theta + \cos^2\theta)}_{=\,1} + 1 + 2\cos\theta = 2 + 2\cos\theta = 2(1 + \cos\theta) \quad \text{(M1)}$

So the whole expression becomes: $\text{LHS} = \frac{2(1 + \cos\theta)}{\sin\theta(1 + \cos\theta)} = \frac{2}{\sin\theta} = \text{RHS} \quad \blacksquare \quad \text{(A1)}$

Where the marks are won and lost

• Choosing to start from the left matters: the right-hand side is already simple, so there is nothing to manipulate down from it. Working down from the busy side is the natural route.
• The $(1 + \cos\theta)^2$ expansion must include the middle term $2\cos\theta$. Writing $1 + \cos^2\theta$ loses the whole proof.
• Spotting the common factor $(1 + \cos\theta)$ in the last step is what makes it cancel. If you multiplied everything out instead, you would be stuck.

Common mistakes

• Working on both sides simultaneously (a logical error the mark scheme penalises).
• Expanding $(1 + \cos\theta)^2$ as $1 + \cos^2\theta$.
• Forgetting to cancel $(1 + \cos\theta)$ at the end and leaving an unsimplified fraction.

Full method: Trig Identities notes. See also the exam technique guide and the Trigonometry pillar.