[4 marks sum]

Question 14 Marks

Prove that $: \tan \left(45^{\circ}- A \right) \tan \left(45^{\circ}+ A \right)=1$.
$\left[\right.$ Hint . $\left.\tan \left(45^{\circ}- A \right)=\tan \left[90^{\circ}-\left(45^{\circ}+ A \right)\right]=\cot \left(45^{\circ}+ A \right)\right]$.

Answer

self

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Question 24 Marks

Prove that : $\sin \left(50^{\circ}+\theta\right)-\cos \left(40^{\circ}-\theta\right)=0$.
$\left[\right.$ Hint . $\left.\sin \left(50^{\circ}+\theta\right)=\sin \left\{90^{\circ}-\left(40^{\circ}-\theta\right)\right\}\right]$

Answer

self

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Question 34 Marks

If $0^{\circ}<\theta<25^{\circ}$, prove that $\cos \left(65^{\circ}+\theta\right)-\sin \left(25^{\circ}-\theta\right)=0$.

Answer

self

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Question 44 Marks

Without using tables, verify that:
$\cos 60^{\circ}=\frac{1-\tan ^2 30^{\circ}}{1+\tan ^2 30^{\circ}}=\frac{1}{2}$.

Answer

self

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Question 54 Marks

Without using tables, verify that:
$\sin 60^{\circ}=\frac{2 \tan 30^{\circ}}{1+\tan ^2 30^{\circ}}=\frac{\sqrt{3}}{2}$.

Answer

self

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Question 64 Marks

Without using tables, verify that:
$\cos 60^{\circ}=\left(\cos ^2 30^{\circ}-\sin ^2 30^{\circ}\right)$.

Answer

self

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Question 74 Marks

If $A=60^{\circ}$ and $B=30^{\circ}$, prove that :
(i) sin (A + B) = sin A cos B + cos A sin B.
(ii) cos (A + B) = cos A cos B - sin A sin B.
(iii) cos (A - B) = cos A cos B + sin A sin B.
(iv) $\tan ( A - B )=\frac{\tan A -\tan B }{1+\tan A \tan B }$.

Answer

self

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Question 84 Marks

If $A=B=45^{\circ}$, show that :
(i) sin (A - B) = sin A cos B - cos A sin B.
(ii) cos (A + B) = cos A cos B - sin A sin B.

Answer

self

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Question 94 Marks

If $A=30^{\circ}$, prove that :
(i) $\sin 2 A=\frac{2 \tan A}{\left(1+\tan ^2 A\right)}$ (ii) $\cos 2 A=\left(\frac{1-\tan ^2 A}{1+\tan ^2 A}\right)$

Answer

self

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Question 104 Marks

If $A =45^{\circ}$, verify that :
(i) sin 2A = 2 sin A cos A
(ii) $\cos 2 A=\left(2 \cos ^2 A-1\right)=\left(1-2 \sin ^2 A\right)$

Answer

self

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Question 114 Marks

If $\sin \theta=\frac{\sqrt{3}}{2}$, find the value of (cosec $\theta$ + cot $\theta$).

Answer

$\sqrt{3}$

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Question 124 Marks

If $\tan \theta=\frac{5}{12}$ and $\theta$ is acute, find the values of sin $\theta$ and cos $\theta$.

Answer

$\sin \theta=\frac{5}{13}, \cos \theta=\frac{12}{13}$

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Question 134 Marks

If $\sin \theta=\frac{3}{5}$ and $\theta$ is an acute angle, find the values of cos $\theta$ and tan $\theta$.

Answer

$\cos \theta=\frac{4}{5}, \tan \theta=\frac{3}{4}$

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Question 144 Marks

If $\operatorname{cosec} \theta=\sqrt{10}$, find the values of other trigonometrical ratios for $\theta$.

Answer

$\sin \theta=\frac{1}{\sqrt{10}}, \cos \theta=\frac{3}{\sqrt{10}}, \tan \theta=\frac{1}{3}, \sec \theta=\frac{\sqrt{10}}{3}, \cot \theta=3.$

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Question 154 Marks

If $\tan \theta=\frac{8}{15}$, find the values of other trigonometrical ratios for $\theta$.

Answer

$\sin \theta=\frac{8}{17}, \cos \theta=\frac{15}{17}, \operatorname{cosec} \theta=\frac{17}{8}, \sec \theta=\frac{17}{15}, \cot \theta=\frac{15}{8}$.

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MATHEMATICS [4 marks sum]

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