Prove the following identities: ^2 ( -1) (1+ ) + ^2 ( -1) (1+ ) =… - Vidyadip

Question

Prove the following identities:
$\frac{\cot^2\theta(\sec\theta-1)}{(1+\sin\theta)}+\frac{\sec^2\theta(\sin\theta-1)}{(1+\sec\theta)}=0$

✓

Answer

$\text{LHS}=\frac{\cot^2\theta(\sec\theta-1)}{(1+\sin\theta)}+\frac{\sec^2\theta(\sin\theta-1)}{(1+\sec\theta)}$
$=\frac{\cot^2\theta(\sec\theta-1)(1+\sec\theta)+\sec^2\theta(\sin\theta-1)(1+\sin\theta)}{(1+\sin\theta)(1+\sec\theta)}$
$=\frac{\cot^2\theta\big(\sec^2\theta-1\big)+\sec^2\theta\big(\sin^2\theta-1\big)}{(1+\sin\theta)(1+\sec\theta)}$
$=\frac{\cot^2\theta\tan^2\theta+\sec^2\theta\big(-\cos^2\theta\big)}{(1+\sin\theta)(1+\sec\theta)}$
$=\frac{\cot^2\theta\tan^2\theta-\sec^2\theta\cos^2\theta}{(1+\sin\theta)(1+\sec\theta)}$
$=\frac{\cot^2\theta\times\frac{1}{\cot^2\theta}-\sec^2\theta\times\frac{1}{\sec^2\theta}}{(1+\sin\theta)(1+\sec\theta)}$
$=\frac{1-1}{(1+\sin\theta)(1+\sec\theta)}$
$=0$
$\therefore\ \text{R.H.S.}=\text{L.H.S.}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from Trigonometric Identities→Trigonometric Identities — all question types→Maths STD 10 question papers→CBSE Board STD 10 question papers→CBSE Board question paper generator→

Similar questions

23 spoons and 17 forks together cost ₹ 1770, while 17 spoons and 23 forks together cost ₹ 1830. Find the cost of a spoon and that of a fork.

A cow is tied with a rope of length 14m at the corner of a rectangular field of dimensions 20m × 16m. Find the area of the field in which the cow can graze.

In an A.P., the first term is 22, $n^{th}$ term is -11 and the sum to first n terms is 66. Find n and d, the common difference.

Write the condition to be satisfied for which equations $ax^2 + 2bx + c = 0$ and $\text{bx}^2-2\sqrt{\text{ac}}\text{x}+\text{b}=0$ have equal roots.

A bag contains cards numbered from 1 to 49. A card is drawn from the bag at random, after mixing the card thoroughly. Find the probability that the number on the drawn card is:

An odd number.
A multiple of 5.
A perfect square.
An even prime number.

2 men and 5 boys can finish a piece of work in 4 days, while 3 men and 6 boys can finish it in 3 days. Find the time taken by one man alone to finish the work and that taken by one boy alone to finish the work.

The $12^{th}$ term of an AP is -13 and the sum of its first four terms is 24. Find the sum of its first 10 terms.

The $24^{\text {th }}$ term of an A.P. is twice its $10^{\text {th }}$ term. Show that its $72^{\text {nd }}$ term is 4 times its $15^{\text {th }}$ term.

For what value of x are the points A(-3, 12), B(7, 6) and C(x, 9) colinear?

The median of the distribution given below is 14.4. Find the values of x and y, if the total frequency is 20.

Class interval:	0-6	6-16	12-18	18-24	24+30
Frequency:	4	x	5	y	1