# Question 4(a):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $\cos x = \cos^2\frac{x}{2}-\sin^2\frac{x}{2}$ or $2\cos^2\frac{x}{2}-1$ or $1-2\sin^2\frac{x}{2}$ | B1 | Any correct appropriate double angle identity for $\cos x$ or possibly $\tan x$ |
| $\Rightarrow \cos x = \left(\frac{1}{\sqrt{1+t^2}}\right)^2-\left(\frac{t}{\sqrt{1+t^2}}\right)^2$ | M1 | Substitutes correct expressions for $\cos\frac{x}{2}$ and/or $\sin\frac{x}{2}$ in terms of $t$ |
| $= \frac{1}{1+t^2}-\frac{t^2}{1+t^2}=\frac{1-t^2}{1+t^2}$ * | A1* | Completes proof with sufficient working shown, no errors; given mark |

# Question 4 (Trigonometry with t-substitution):

## Part (a):

| Working | Mark | Guidance |
|---------|------|----------|
| $= \frac{2t}{1+t^2} \times \frac{1-t^2}{2t} = \frac{1-t^2}{1+t^2}$ | M1 | Substituting correct expressions for $\cos\frac{x}{2}$, $\sin\frac{x}{2}$ and $\tan\frac{x}{2}$ in terms of $t$ |
| Completion to $\frac{1-t^2}{1+t^2}$ | A1* | Completing the proof with sufficient working shown, no errors. Allow $\theta$ for $x$ for first 2 marks but must obtain $\cos x = \ldots$ for A1* |

**Special Case** (quoting $\tan x$ and/or $\sin x$ to verify): Scores SC B1 only

## Part (b):

| Working | Mark | Guidance |
|---------|------|----------|
| $3\tan x - 10\cos x = 10 \Rightarrow 3 \times \frac{2t}{1-t^2} - 10 \times \frac{1-t^2}{1+t^2} = 10$ | B1 | Applies part (a) and half angle formula for $\tan$ to give correct equation in $t$ |
| $\Rightarrow 6t(1+t^2) - 10(1-t^2)^2 = 10(1-t^2)(1+t^2)$ leading to $6t^3 + 20t^2 + 6t - 20 = 0$ or $3t^3 + 10t^2 + 3t - 10 = 0$ | M1 | Attempts to multiply by $1-t^2$ and $1+t^2$, expands and simplifies to a cubic in $t$ |
| $\Rightarrow (t+2)(\text{"6"}t^2 + \ldots t + \ldots) = 0$ | M1 | Attempts to take factor $(t+2)$ out of cubic. May be done by inspection or long division; must obtain correct coefficient for $t^2$ and reach a 3-term quadratic |
| $\Rightarrow (t+2)(6t^2+8t-10)=0$ or $(t+2)(3t^2+4t-5)=0$ | A1 | Correct equation. Must see equation written down not just values for $a$, $b$ and $c$ |

## Part (c):

| Working | Mark | Guidance |
|---------|------|----------|
| $t+2=0 \Rightarrow x = 2\times\arctan(-2) = \ldots$ | M1 | Attempts to solve $t+2=0$, look for attempt at $\arctan(\pm 2)$ and attempt to double. Allow in radians |
| $x = -126.86\ldots°$ (allow awrt $-127°$) | A1 | awrt $-127°$ |
| $3t^2+4t-5=0 \Rightarrow t = \frac{-4 \pm \sqrt{16-4\times3\times-5}}{6} = \frac{-2\pm\sqrt{19}}{3}$ $(= 0.786\ldots, -2.11\ldots)$ $\Rightarrow x = 2\times\arctan(\ldots) = \ldots$ | M1 | Solves quadratic and attempts arctan and doubles. Usual rules apply for solving quadratic |
| $x = \text{(awrt)} -129°, 76.4°$ $(-129.486\ldots, 76.355\ldots)$ | A1 (one), A1 (both) | Either awrt $-129°$ or awrt $76.4°$ for first A1; both awrt $-129°$ and awrt $76.4°$ for second A1. No other values in range. Values outside range can be ignored |

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