Additional Mathematics 4037/13 — October/November 2010

Walkthrough

The task is to prove an identity: start from one side and derive the other exactly. The left-hand side contains $\sec x$, which is not defined directly in terms of sine and cosine, so the first move is to rewrite it as $\frac{1}{\cos x}$ — this is what earns the first method mark for 'dealing with sec and fractions'.

With both terms now over cosine, they are combined into a single fraction with common denominator $\cos x$:

The numerator $1 - \cos^2 x$ is exactly where the Pythagorean identity $\sin^2 x + \cos^2 x = 1$ applies — this substitution earns the second method mark for 'use of trig identity'. Replacing $1 - \cos^2 x$ by $\sin^2 x$ gives $\frac{\sin^2 x}{\cos x}$, which splits as $\sin x \times \frac{\sin x}{\cos x}$, and since $\tan x = \frac{\sin x}{\cos x}$ the expression becomes $\sin x \tan x$, matching the right-hand side and earning the accuracy mark.

Key Takeaways

• An identity proof works from one side to the other; never assume the result and manipulate both sides simultaneously.
• Convert everything to sines and cosines first: $\sec x = \frac{1}{\cos x}$, $\tan x = \frac{\sin x}{\cos x}$.
• Recognise difference-of-squares patterns like $1 - \cos^2 x$ as invitations to use $\sin^2 x + \cos^2 x = 1$.
• Show every algebraic line — schemes for 'show that' parts require sufficient correct detail.

Common Mistakes

• Starting from the target $\sin x \tan x$ and working backwards; proofs must be derived forward from the given side (AG parts).
• Writing $\sec x - \cos x = \tan x \sin x$ immediately without any intermediate working — no marks are awarded for unsupported steps.
• Misquoting the identity, e.g. writing $1 - \cos^2 x = \cos^2 x - 1$ or confusing it with $1 + \tan^2 x = \sec^2 x$.
• Cancelling incorrectly, such as dividing numerator and denominator by $\sin x$ when the denominator is $\cos x$.
• Skipping the final recognition step $\frac{\sin x}{\cos x} = \tan x$, leaving the answer as $\frac{\sin^2 x}{\cos x}$ which does not match the printed target.

Things to Be Careful About

• This is an AG ('Answer Given') part: the target is printed, so you must show sufficient correct detail on every line between the given expression and the target — each of the three mark scheme steps must appear explicitly.
• Keep all trigonometric functions upright ($\sec x$, $\cos x$, $\sin x$, $\tan x$) and write the argument $x$ consistently.
• No numerical accuracy issues arise here, but the final line must be exactly $\sin x \tan x$ (or an equivalent form accepted as 'oe').