University Maths Solution
Maths Question | |
---|---|
Question 1 |
$\text{Establish }(\tan \theta +\cot \theta )\tan \theta ={{\sec }^{2}}\theta $ |
Question 2 |
$\text{Establish }\frac{\sin t}{1-\cos t}=\csc t+\cot t$ |
Question 3 |
$\text{Establish }\cot \alpha (\sec \alpha +\cos ec\alpha )$ |
Question 4 |
$\text{Establish }\frac{1}{\sin \alpha +\cos \alpha }=\frac{\tan \alpha +\cot \alpha }{\sec \alpha +\operatorname{cosec}\alpha }$ |
Question 5 |
$\text{Establish the identity }\frac{1-\cos \alpha }{\sin \alpha }=\frac{1}{\operatorname{cosec}\alpha +\cot \alpha }$ |
Question 6 |
$\text{Establish that }\frac{{{\tan }^{2}}\theta +{{\cos }^{2}}\theta }{\sec \theta +\sin \theta }=\sec \theta -\sin \theta$ |
Question 7 |
$\text{Establish that }\sqrt{\frac{1-\sin x}{1+\sin x}}=\frac{\cos x}{1+\sin x}$ |
Question 8 |
$\text{Establish that }\frac{1-\sin \theta +\cos \theta }{1-\sin \theta }=\frac{1+\sin \theta +\cos \theta }{\cos \theta }$ |
Question 9 |
$\text{Establish that }\frac{\sin \theta }{1+\cos \theta }+\frac{1+\cos \theta }{\sin \theta }=2\operatorname{cosec}\theta $ |
Question 10 |
$\text{Establish that }\frac{\tan u-\tan v}{1+\tan u\tan v}=\frac{\cot v-\cot u}{1+\cot u\cot v}$ |
Question 11 |
$\text{Establish that }\frac{1}{\tan \beta +\cot \beta }=\sin \beta \cos \beta $ |
Question 12 |
$\text{Find the values in terms of }p\text{ and }q\text{ of }\frac{p\cos \theta +q\sin \theta }{p\cos \theta -q\sin \theta }\text{ where }\cot \theta =\frac{p}{q}$ |
Question 13 |
$\text{If }\cos \theta =a,\text{ find the value of }\operatorname{cosec}\left( \tfrac{\pi }{2}+\theta \right)\text{ and }\sin \left( \tfrac{3\pi }{2}-\theta \right)$ |
Question 14 |
$\begin{align} & \text{if }\cos \theta =\tfrac{4}{5}\text{ (i) }\sin \theta \text{ (ii) }\operatorname{cosec}\theta \text{ (iii) }\theta \\ & \text{If }\sin \theta =-\tfrac{12}{13}\text{ (i) cos}\theta \text{ (ii) }\tan \theta \\\end{align}$ |
Question 15 |
$\begin{align} & \text{Write down the general solution of the trigonometric equation} \\ & \sin (\theta +{{45}^{\circ }})=\frac{1}{2} \\\end{align}$ |
Question 16 |
$\begin{align} & \text{Write down the general solution of the trigonometric equations} \\ & \cos (2\theta -{{30}^{\circ }})=0 \\\end{align}$ |
Question 17 |
$\begin{align} & \text{Write down the general solution of the trigonometric equations} \\ & \tan 3\theta =\sqrt{3} \\\end{align}$ |
Question 18 |
$\begin{align} & \text{Write down the general solution of the trigonometric equations} \\ & 2\sin 2\theta +1=0 \\\end{align}$ |
Question 19 |
$\begin{align} & \text{Write down the general solution of the trigonometric equations} \\ & 2\cos ({{90}^{\circ }}-\theta )=\sqrt{2} \\\end{align}$ |
Question 20 |
$\text{Verify }\cos \left( \theta +\frac{\pi }{4} \right)=\frac{\sqrt{2}}{2}(\cos \theta -\sin \theta )$ |
Question 21 |
$\text{Verify }\operatorname{cosec}\theta \left( \frac{\pi }{4}-u \right)=\sec u$ |
Question 22 |
$\text{Verify }\cot \left( t-\frac{\pi }{3} \right)=\frac{\sqrt{3}(\tan t+3)}{\tan t-\sqrt{3}}$ |
Question 23 |
$\text{Verify }\sec 2m=\frac{{{\sec }^{2}}m}{2-{{\sec }^{2}}m}$ |
Question 24 |
$\begin{align} & \text{Obtain the first general solution of the equation and then use the general } \\ & \text{solution to find all solution in the range }{{0}^{\circ }}\le \theta \le {{360}^{\circ }} \\ & 8{{\sin }^{2}}\theta -6\cos \theta =3 \\\end{align}$ |
Question 25 |
$\begin{align} & \text{Obtain the first general solution of the equation and then use the general } \\ & \text{solution to find all solution in the range }{{0}^{\circ }}\le \theta \le {{360}^{\circ }} \\ & 7{{\sec }^{2}}\theta =6\tan \theta +8 \\\end{align}$ |
Question 26 |
$\begin{align} & \text{Obtain the first general solution of the equation and then use the general } \\ & \text{solution to find all solution in the range }{{0}^{\circ }}\le \theta \le {{360}^{\circ }} \\ & 3\sin 3\theta -\operatorname{cosec}3\theta +2=0 \\\end{align}$ |
Question 27 |
$\begin{align} & \text{Obtain the first general solution of the equation and then use the general } \\ & \text{solution to find all solution in the range }{{0}^{\circ }}\le \theta \le {{360}^{\circ }} \\ & 2{{\cos }^{2}}\theta =\sin \theta +1 \\\end{align}$ |