Prove the following trigonometric identities. ^2A( -1) 1+ = ^2A (… - Vidyadip

Question

Prove the following trigonometric identities.
$\frac{\cot^2\text{A}(\sec\text{A}-1)}{1+\sin\text{A}}=\sec^2\text{A}\Big(\frac{1-\sin\text{A}}{1+\sin\text{A}}\Big)$

✓

Answer

We have to prove $\frac{\cot^2\text{A}(\sec\text{A}-1)}{1+\sin\text{A}}=\sec^2\text{A}\Big(\frac{1-\sin\text{A}}{1+\sin\text{A}}\Big)$
We know that, $\sin^2\text{A}+\cos^2\text{A}=1$
So,
$\text{L.H.S}=\frac{\cot^2\text{A}(\sec\text{A}-1)}{1+\sin\text{A}}=\sec^2\text{A}\Big(\frac{1-\sin\text{A}}{1+\sin\text{A}}\Big)$
$=\frac{\frac{\cos^2\text{A}}{\sin^\text{2}\text{A}}\big(\frac{1}{\cos\text{A}}-1\Big)}{1+\sin\text{A}}$
$=\frac{\frac{\cos^2\text{A}}{\sin^2\text{A}}\frac{1-\cos\text{A}}{\cos\text{A}}}{1+\sin\text{A}}$
$=\frac{\cos\text{A}(1-\cos\text{A})}{\sin^2\text{A}(1+\sin\text{A})}$
$=\frac{\cos\text{A}(1-\cos\text{A})}{(1-\cos^2\text{A})(1+\sin\text{A})}$
$=\frac{\cos\text{A}(1-\cos\text{A})}{(1-\cos\text{A})(1+\cos\text{A})(1+\sin\text{A})}$
$=\frac{\cos\text{A}}{(1+\cos\text{A})(1+\sin\text{A})}$
$=\frac{\frac{1}{\sec\text{A}}}{\Big(1+\frac{1}{\sec\text{A}}\Big)(1+\sin\text{A})}$
$=\frac{\frac{1}{\sec\text{A}}}{\Big(\frac{\sec\text{A}+1}{\sec\text{A}}\Big)(1+\sin\text{A})}$
$=\frac{1}{(\sec\text{A}+1)(1+\sin\text{A})}$
Multiplying both the numerator ans denominator by $(1-\sin\text{A})$ we have
$=\frac{(1-\sin\text{A})}{(\sec\text{A}+1)(1+\sin\text{A})(1-\sin\text{A})}$
$=\frac{1-\sin\text{A}}{(\sec\text{A}+1)\cos^2\text{A}}$
$=\frac{(1-\sin\text{A})}{(\sec\text{A}+1)\cos^2\text{A}}$
$=\sec^2\text{A}\frac{(1-\sin\text{A})}{(\sec\text{A}+1)}$
$=\sec^2\text{A}\Big(\frac{1-\sin\text{A}}{1+\sec\text{A}}\Big)=\text{R.H.S}$
$\therefore\ \text{L.H.S}=\text{R.H.S}$

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 "4 Marks Questions" from Trigonometric Identities→Trigonometric Identities — all question types→Maths STD 10 question papers→Maharashtra Board STD 10 question papers→Maharashtra Board question paper generator→

Similar questions

Evaluate the following:
If $\sin(\text{A}+\text{B})=1$ and $\cos(\text{A}-\text{B})=1,0^\circ\leq(\text{A}+\text{B})\leq90^\circ$ and $\text{A}>\text{B}$ then find A and B.

By actual division, show that $x^2- 3$ is a factor of $2x^4+ 3x^3- 2x^2- 9x - 12$

Find how many intergers between $200$ and $500$ are divisible by $8$.

On selling a T.V. at 5% gain and a fridge at 10% gain, a shopkeeper gains Rs. 2000. But if he sells the T.V. at 10% gain the fridge at 5% loss. He gains Rs. 1500 on the transaction. Find the actual prices of T.V. and fridge.

Find the mean of each of the following frequency distributions:

Class interval	0-6	6-12	12-18	18-24	24-30
Frequency	7	5	10	12	6

A circus tent has cylindrical shape surmounted by a conical roof. The radius of the cylindrical base is $20m$. The heights of the cylindrical and conical portions are $4.2m$ and $2.1m$ respectively. Find the volume of the tent.

In a rectangle, if the length is increased by 3 meters and breadth is decreased by 4 meters, the area of the rectangle is reduced by 67 square meters. If length is reduced by 1 meter and breadth is increased by 4 meters, the area is increased by 89 Sq. meters. Find the dimensions of the rectangle.

Solve the following systems of equations:
$\frac{\text{x}}{3}+\frac{\text{y}}{4}=11,$
$\frac{5\text{x}}{6}-\frac{\text{y}}{3}=-7.$

Find the mean of each of the following frequency distributions:

Classes	$25-29$	$30-34$	$35-39$	$40-44$	$45-49$	$50-54$	$55-59$
Frequency	$14$	$22$	$16$	$6$	$5$	$3$	$4$

Candidate of four schools appear in a mathematics test. The data were as follows:

Schools	No. of Candidates	Average Score
i	60	75
ii	48	80
iii	Not available	55
iv	40	50

If the average score of the candidates of all the four schools is 66, find the number of candidates that appeared from school III.