Mathematics 9709/33 — May/June 2017

Walkthrough

We need to prove an identity, so we start with the left-hand side and transform it until it equals the right-hand side.

First, rewrite $\cot x$ and $\tan x$ using the basic definitions:

$$\cot x = \frac{\cos x}{\sin x}, \qquad \tan x = \frac{\sin x}{\cos x}.$$

This is the first mark: the expression is now written entirely in terms of $\cos x$ and $\sin x$.

Substitute these into the numerator and denominator. Both become fractions. To combine them, give each fraction the common denominator $\sin x \cos x$:

$$\cot x - \tan x = \frac{\cos^2 x - \sin^2 x}{\sin x \cos x}$$

and

$$\cot x + \tan x = \frac{\cos^2 x + \sin^2 x}{\sin x \cos x}.$$

Dividing the first fraction by the second cancels the common denominator $\sin x \cos x$, leaving

$$\frac{\cos^2 x - \sin^2 x}{\cos^2 x + \sin^2 x}.$$

Now use the Pythagorean identity $\cos^2 x + \sin^2 x = 1$. This is the second mark. The denominator becomes 1, so the expression simplifies to

$$\cos^2 x - \sin^2 x.$$

Finally, recognise that $\cos^2 x - \sin^2 x$ is the double-angle formula for $\cos 2x$. This gives the required result and earns the final mark.

Key Takeaways

• The definitions $\tan x = \frac{\sin x}{\cos x}$ and $\cot x = \frac{\cos x}{\sin x}$ allow many trig expressions to be rewritten in terms of $\sin x$ and $\cos x$.
• The Pythagorean identity $\cos^2 x + \sin^2 x = 1$ is often used to simplify denominators or numerators.
• Recognising $\cos^2 x - \sin^2 x = \cos 2x$ is essential for proving double-angle identities.
• When simplifying a compound fraction, combine the numerator and denominator separately, then cancel common factors.

Common Mistakes

• Dividing fractions incorrectly: $\frac{a/b}{c/b} = \frac{a}{c}$, not $\frac{a}{b} \cdot \frac{c}{b}$.
• Sign errors: the numerator must be $\cos^2 x - \sin^2 x$, not $\sin^2 x - \cos^2 x$.
• Writing $\cos^2 x + \sin^2 x = 0$ or replacing it by something other than 1.
• Stopping at $\cos^2 x - \sin^2 x$ without applying the double-angle formula.
• The mark scheme requires method: an unsupported final answer would not earn full marks.

Things to Be Careful About

• The identity is only defined where both $\tan x$ and $\cot x$ are defined, i.e. where $\cos x \neq 0$ and $\sin x \neq 0$.
• Keep the common denominator $\sin x \cos x$ until it cancels; do not cancel it too early.
• Use exactly $\cos^2 x + \sin^2 x = 1$; this is the Pythagorean identity needed for the M1 mark.
• The final step must state $\cos 2x$ to earn the A1 mark.