Question
Prove.
$(\cot A-\operatorname{cosec} A)^2=\frac{1-\cos A}{1+\cos A}$
$(\cot A-\operatorname{cosec} A)^2=\frac{1-\cos A}{1+\cos A}$
✓
Answer
$\text { R.H.S }=\frac{1-\cos A}{1+\cos A}$
$ =\frac{(1-\cos A)(1-\cos A)}{(1+\cos A)(1-\cos A)}$
v =\frac{(1-\cos A)^2}{1-\cos ^2 A} $
$=\frac{(1-\cos A)^2}{\sin ^2 A} $
$ =\left(\frac{1}{\sin A}-\frac{\cos A}{\sin A}\right)^2$
$=(\operatorname{cosec} A-\cot A)^2$
$=(\cot A-\operatorname{cosec} A)^2 $
$ =\text { L.H.S }$
$ =\frac{(1-\cos A)(1-\cos A)}{(1+\cos A)(1-\cos A)}$
v =\frac{(1-\cos A)^2}{1-\cos ^2 A} $
$=\frac{(1-\cos A)^2}{\sin ^2 A} $
$ =\left(\frac{1}{\sin A}-\frac{\cos A}{\sin A}\right)^2$
$=(\operatorname{cosec} A-\cot A)^2$
$=(\cot A-\operatorname{cosec} A)^2 $
$ =\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
If the points $A (6,1), B (8,2), C (9,4)$ and $D (p, 3)$ are the vertices of a parallelogram, taken in order, find the value of $p$.
→If $\frac{\cos A}{\cos B}=m$ and $\frac{\cos A}{\sin B}= n$ show that: $\left(m^2+n^2\right) \cos ^2 B=n^2$.
→Two dice (each bearing numbers 1 to 6) are rolled together. Find the probability that the sum of the numbers on the upper-most faces of two dice is: less than 6
Prove that $(5 x+4)$ is a factor of $5 x^3+4 x^2-5 x-4$. Hence factorize the given polynomial completely.
→In the given figure, O is the centre of the circle. Tangents at A and B meet at C. If angle ACO = 30°, find: angle AOB
Akash, an employee of a bank, has a saving bank account in his bank that pays him interest at the rate of 5% p.a., which is compounded every June and December. His passbook entries are as follow :
Calculate the interest due at the end of June and find the balance on July 1, if he deposits a cash of? 100 on July 1, which is also entered immediately.
| Date | Particulars | Withdrawals(₹) | Deposits(₹) | Balance(₹) |
| Feb. 3, 1981 | By cash | - | 500·00 | 500·00 |
| Feb, 11 | To cheque no. 371 | 200·00 | - | 300·00 |
| Feb. 11 | By cheque | - | 700·00 | 1,000·00 |
| March 1 | By salary | - | 2,350·00 | 3,350·00 |
| March 4 | To withdrawals slip | 1,500·00 | - | 1,850·00 |
| March 31 | To Urnil | 150·00 | - | 1,700·00 |
| April 1 | By salary | - | 2.350·00 | 4,050·00 |
| April 2 | To Sri Ram | 1,800·00 | - | 2,250·00 |
| May 1 | By salary | - | 2,350·00 | 4,600·00 |
| May 3 | To accountant | 2,000·00 | - | 2,600·00 |
BC is a chord of a circle with centre $\mathrm{O} . \mathrm{A}$ is a point on major arc BC as shown in the figure. Prove that $\angle \mathrm{BAC}+\angle \mathrm{OBC}=90^{\circ}$.
→On a graph paper, draw the line x = 6. Now, on the same graph paper, draw the locus of the point which moves in such a way that its distantce from the given line is always equal to 3 units
Given:
A = {x: 11x – 5 > 7x + 3, x ∈ R} and
B = {x: 18x – 9 ≥ 15 + 12x, x ∈ R}.
Find the range of set A ∩ B and represent it on the number line.
A = {x: 11x – 5 > 7x + 3, x ∈ R} and
B = {x: 18x – 9 ≥ 15 + 12x, x ∈ R}.
Find the range of set A ∩ B and represent it on the number line.
If $\left[\begin{array}{cc}1 & 4 \\ -2 & 3\end{array}\right]+2 M=3\left[\begin{array}{cc}3 & 2 \\ 0 & -3\end{array}\right]$, find the matrix $M$.
→