CAIE IGCSE Additional Math 10.4 Use Trigonometric Identities Questions

$\frac{\sin \theta}{\sqrt{\cot ^{2} \theta}}+\frac{1}{\sqrt{\sec ^{2} \theta}}$
for correct use of the trigonometry
identity
$\operatorname{cosec}^{2} \theta-1=\cot ^{2} \theta$$\sin \theta \tan \theta+\cos \theta$ oe
$\sin \theta \times \frac{\sin \theta}{\cos \theta}+\cos \theta$$\frac{\sin ^{2} \theta+\cos ^{2} \theta}{\cos \theta}=\frac{1}{\cos \theta}=\sec \theta$
Dep on all 3 previous marks
Last mark is not available if persistent
missing of $\theta$

Alternative

$\frac{\sin \theta}{\sqrt{\cot ^{2} \theta}}+\frac{1}{\sqrt{\sec ^{2} \theta}}$
for correct use of the trigonometry
identity
$\operatorname{cosec}^{2} \theta-1=\cot ^{2} \theta$$\frac{\sec \theta \sin \theta+\cot \theta}{\cot \theta \sec \theta}$$\frac{\frac{\sin \theta}{\cos \theta}+\frac{\cos \theta}{\sin \theta}}{\frac{\cos \theta}{\sin \theta} \times \frac{1}{\cos \theta}}$$$\frac{\frac{\sin ^{2} \theta+\cos ^{2} \theta}{\cos \theta \sin \theta}}{\frac{1}{\sin \theta}} \text { leading to } \frac{1}{\cos \theta}$$
Dep on all 3 previous marks
Last mark is not available if persistent
missing $\theta$

$\cos x=\frac{1}{\alpha}$
could be implied by $\frac{1}{\alpha^{2}}=1-\sin ^{2} x$$\sin ^{2} x+\left(\frac{1}{\alpha}\right)^{2}=1$ oe

OR [opposite ${ }^{2}=$ ] $\alpha^{2}-1$ soi and use of $\sin x=\frac{\text { opposite }}{\text { hypotenuse }}$ at some stage leading to $\sin x=[ \pm] \sqrt{1-\frac{1}{\alpha^{2}}}$ oe
$\sin x=-\sqrt{1-\frac{1}{\alpha^{2}}}$ oe