Question
Prove the following identities:
$\big(1+\tan^2\theta\big)\big(1+\cot^2\theta\big)=\frac{1}{\big(\sin^2\theta-\sin^4\theta\big)}$
$\big(1+\tan^2\theta\big)\big(1+\cot^2\theta\big)=\frac{1}{\big(\sin^2\theta-\sin^4\theta\big)}$
✓
Answer
$\text{L.H.S.}=\big(1+\tan^2\theta\big)\big(1+\cot^2\theta\big)$
$=\sec^2\theta\text{ cosec}^2\theta$
$=\frac{1}{\sin^2\theta\cos^2\theta}=\frac{1}{\sin^2\theta\big(1-\sin^2\theta\big)}$
$=\frac{1}{\sin^2\theta-\sin^4\theta}$
$=\text{R.H.S.}$
$\therefore\text{R.H.S.}=\text{L.H.S.}$
$=\sec^2\theta\text{ cosec}^2\theta$
$=\frac{1}{\sin^2\theta\cos^2\theta}=\frac{1}{\sin^2\theta\big(1-\sin^2\theta\big)}$
$=\frac{1}{\sin^2\theta-\sin^4\theta}$
$=\text{R.H.S.}$
$\therefore\text{R.H.S.}=\text{L.H.S.}$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
100 surnames were randomly picked up from a local telephone directory and the frequency distribution of the number of letters in the English alphabets in the surnames was obtained as follows:
Determine the median number of letters in the surnames. Find the mean number of letters in the surnames. Also, and the modal size of the surnames.
|
Number of letters
|
1-4
|
4-7
|
7-10
|
10-13
|
13-16
|
16-19
|
|
Number surnames
|
6
|
30
|
40
|
16
|
4
|
4
|
Solve for x and y:$\frac{\text{bx}}{\text{a}}+\frac{\text{ay}}{\text{b}}=\text{a}^2+\text{b}^2,$
$\text{x}+\text{y}=\text{2ab}$
→$\text{x}+\text{y}=\text{2ab}$
Two solid cones A and B are placed in a cylindrical tube as shown in the figure. The ratio of their capacitites are 2 : 1. Find the heights and capacities of the cones. Also, find the volume of the remaining portion of the cylinder.
Find the sum of first 51 terms of an AP whose second and third terms are 14 and 18 respectively.
The vertices of ΔABC are (-2, 1), (5, 4) and (2, -3) respectively. Find the area of the triangle and the length of the altitude through A.
A chord of a circle of radius 12 cm subtends an angle of $120^o$ at the center. Find the area of the corresponding segment of the circle. (Using $\pi$ = 3.14 and $\sqrt3$ = 1.73)
→Prove that $5\sqrt2$ is an irrational number.
→Short-Answer Question:
If $\alpha$ and $\beta$ are the zeroes of a polynomial $f(x) = x^2 - 5x + k$, such that $\alpha-\beta=1$ find the value of k.
→If $\alpha$ and $\beta$ are the zeroes of a polynomial $f(x) = x^2 - 5x + k$, such that $\alpha-\beta=1$ find the value of k.
A life insurance agent found the following data for distribution of ages of 100 policy holders. Calculate the median age, if policies are only given to persons having age 18 years onwards but less than 60 year.
| Age (in years) | Number of policyholders |
| Below 20 | 2 |
| Below 25 | 6 |
| Below 30 | 24 |
| Below 35 | 45 |
| Below 40 | 78 |
| Below 45 | 89 |
| Below 50 | 92 |
| Below 55 | 98 |
| Below 60 | 100 |
Solve the following system of equations by the method of cross-multiplication:
$\frac{\text{x}}{\text{b}}=\frac{\text{y}}{\text{b}},$
$\text{ax}+\text{by}=\text{a}^2+\text{b}^2.$
→$\frac{\text{x}}{\text{b}}=\frac{\text{y}}{\text{b}},$
$\text{ax}+\text{by}=\text{a}^2+\text{b}^2.$