2. (a) Use the identity \(\cos ^ { 2 } \theta + \sin ^ { 2 } \theta = 1\) to prove that \(\tan ^ { 2 } \theta = \sec ^ { 2 } \theta - 1\).
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Question 2:
Part (a)
Answer Marks
Guidance
Answer/Working Mark
Guidance
\(\cos^2\theta + \sin^2\theta = 1 \div \cos^2\theta\): \(\frac{\cos^2\theta}{\cos^2\theta} + \frac{\sin^2\theta}{\cos^2\theta} = \frac{1}{\cos^2\theta}\) M1
Dividing \(\cos^2\theta + \sin^2\theta = 1\) by \(\cos^2\theta\) to give underlined equation
\(1 + \tan^2\theta = \sec^2\theta\); \(\tan^2\theta = \sec^2\theta - 1\) AG A1 cso
Complete proof. No errors seen
Part (b)
Answer Marks
Guidance
Answer/Working Mark
Guidance
\(2(\sec^2\theta - 1) + 4\sec\theta + \sec^2\theta = 2\) M1
Substituting \(\tan^2\theta = \sec^2\theta - 1\) into eqn* to get a quadratic in \(\sec\theta\) only
\(3\sec^2\theta + 4\sec\theta - 4 = 0\) M1
Forming a three term "one sided" quadratic expression in \(\sec\theta\)
\((\sec\theta + 2)(3\sec\theta - 2) = 0\) M1
Attempt to factorise or solve a quadratic
\(\cos\theta = -\frac{1}{2}\) or \(\cos\theta = \frac{3}{2}\) A1
\(\cos\theta = -\frac{1}{2}\)
\(\theta_1 = 120°\) A1
\(120°\)
\(\theta_2 = 240°\) B1\(\checkmark\)
\(240°\) or \(\theta_2 = 360° - \theta_1\) when solving using \(\cos\theta = ...\)
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# Question 2:
## Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\cos^2\theta + \sin^2\theta = 1 \div \cos^2\theta$: $\frac{\cos^2\theta}{\cos^2\theta} + \frac{\sin^2\theta}{\cos^2\theta} = \frac{1}{\cos^2\theta}$ | M1 | Dividing $\cos^2\theta + \sin^2\theta = 1$ by $\cos^2\theta$ to give underlined equation |
| $1 + \tan^2\theta = \sec^2\theta$; $\tan^2\theta = \sec^2\theta - 1$ **AG** | A1 cso | Complete proof. No errors seen |
## Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $2(\sec^2\theta - 1) + 4\sec\theta + \sec^2\theta = 2$ | M1 | Substituting $\tan^2\theta = \sec^2\theta - 1$ into eqn* to get a quadratic in $\sec\theta$ only |
| $3\sec^2\theta + 4\sec\theta - 4 = 0$ | M1 | Forming a three term "one sided" quadratic expression in $\sec\theta$ |
| $(\sec\theta + 2)(3\sec\theta - 2) = 0$ | M1 | Attempt to factorise or solve a quadratic |
| $\cos\theta = -\frac{1}{2}$ or $\cos\theta = \frac{3}{2}$ | A1 | $\cos\theta = -\frac{1}{2}$ |
| $\theta_1 = 120°$ | A1 | $120°$ |
| $\theta_2 = 240°$ | B1$\checkmark$ | $240°$ or $\theta_2 = 360° - \theta_1$ when solving using $\cos\theta = ...$ |
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2. (a) Use the identity $\cos ^ { 2 } \theta + \sin ^ { 2 } \theta = 1$ to prove that $\tan ^ { 2 } \theta = \sec ^ { 2 } \theta - 1$.\\
(b) Solve, for $0 \leqslant \theta < 360 ^ { \circ }$, the equation
$$2 \tan ^ { 2 } \theta + 4 \sec \theta + \sec ^ { 2 } \theta = 2$$
\hfill \mbox{\textit{Edexcel C3 2009 Q2 [8]}}