# Practice Questions – Trigonometric Identities and Equations

1. (i) Prove the identity  $\left(\mathrm{sin}\theta +\mathrm{cos}\theta \right)\left(1–\mathrm{sin}\theta \mathrm{cos}\theta \right)={\mathrm{sin}}^{3}\theta –{\mathrm{cos}}^{3}\theta$ .   [3 marks]

(ii) Hence solve the equation $\left(\mathrm{sin}\theta +\mathrm{cos}\theta \right)\left(1–\mathrm{sin}\theta \mathrm{cos}\theta \right)=3{\mathrm{cos}}^{3}\theta$ , for  $0°\le \theta \le 360°$ .   [3 marks]

2. (i) Express  in the form $R\mathrm{cos}\left(\theta +\alpha \right)$ , where $R>0$ and  $0<\alpha <\frac{\pi }{2}$ . Give the value of  $\alpha$ correct to 4 decimal places. [3 marks]

(ii) Using your answer from part (i), solve the equation  for  $0<\theta <2\pi$ .   [5 marks]

3. (i) Prove the identity  $\frac{1+\mathrm{cos}\theta }{\mathrm{sin}\theta }+\frac{\mathrm{sin}\theta }{1+\mathrm{cos}\theta }=\frac{2}{\mathrm{sin}\theta }$ . [3 marks]

(ii) Hence solve the equation  $\frac{1+\mathrm{cos}\theta }{\mathrm{sin}\theta }+\frac{\mathrm{sin}\theta }{1+\mathrm{cos}\theta }=\frac{3}{\mathrm{cos}\theta }$ for  $0°\le \theta \le 360°$ . [4 marks]

4. (i) Express  in the form $R\mathrm{cos}\left(\theta –\alpha \right)$ where R > 0 and $0°<\alpha <90°$ , giving the value of  $\alpha$ correct to 2 decimal places. [3 marks]

(ii) Hence solve the equation  for  $0°<\theta <360°$ .   [4 marks]