Question
Prove the following identities:
$1+\frac{\cot^2\theta}{(1+\text{cosec }\theta)}=\text{sec}\theta$
$1+\frac{\cot^2\theta}{(1+\text{cosec }\theta)}=\text{sec}\theta$
✓
Answer
$\text{L.H.S.}=1+\frac{\cot^2\theta}{(1+\text{cosec }\theta)}$
$=1+\frac{\text{cosec}^2\theta-1}{(1+\text{cosec}\theta)}$
$=\frac{1+\text{cosec}+\text{cosec}^2\theta-1^2}{1+\text{cosec}\theta}$
$=\frac{(1+\text{cosec}\theta)+(\text{cosec}\theta+1)(\text{cosec}\theta-1)}{(1+\text{cosec}\theta)}$
$=\frac{(1+\text{cosec}\theta)\big[1+\text{cosec}\theta-1)\big]}{(1+\text{cosec}\theta)}$
$=1+\text{cosec}\theta-1$
$=\text{cosec}\theta$
$=\text{R.H.S.}$
Hence, LHS = RHS.
$=1+\frac{\text{cosec}^2\theta-1}{(1+\text{cosec}\theta)}$
$=\frac{1+\text{cosec}+\text{cosec}^2\theta-1^2}{1+\text{cosec}\theta}$
$=\frac{(1+\text{cosec}\theta)+(\text{cosec}\theta+1)(\text{cosec}\theta-1)}{(1+\text{cosec}\theta)}$
$=\frac{(1+\text{cosec}\theta)\big[1+\text{cosec}\theta-1)\big]}{(1+\text{cosec}\theta)}$
$=1+\text{cosec}\theta-1$
$=\text{cosec}\theta$
$=\text{R.H.S.}$
Hence, LHS = RHS.
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
A solid cuboid of iron with dimensions 53cm × 40cm × 15cm is melted and recast into a cylindrical pipe. The outer and inner diameters of pipe are 8cm and 7cm respectively. Find the length of pipe.
Show that the following numbers are irrational.
$6+\sqrt{2}$
→$6+\sqrt{2}$
If the points A(-1, -4), B(b, c) and C(5, -1) are collinear and 2b + c = 4, find the values of b and c.
Solve : x² + x + 5 = 0
Find:
Is $68$ a term of the A.P. $7, 10, 13, ...?$
→Is $68$ a term of the A.P. $7, 10, 13, ...?$
In the given figure, DE || BC such that AD = x cm, DB = (3x + 4)cm, AE = (x + 3)cm and EC = (3x + 19)cm. Find the value of x.
how many two-digit numbers are divisible by 5?
Activity :- Two-digit numbers divisible by 5 are, 10,15,20........95.
Here, d=5, therefore this sequence is an A.P.
Here a=10, d=5, Tn=95, n=?
$t _{ n }$= a+(n-1____)
____= 10+(n-1)×5
____= (n-1)×5
____= (n-1)
therefore n=____
there are ____two-digit numbers divisible by 5.
→Activity :- Two-digit numbers divisible by 5 are, 10,15,20........95.
Here, d=5, therefore this sequence is an A.P.
Here a=10, d=5, Tn=95, n=?
$t _{ n }$= a+(n-1____)
____= 10+(n-1)×5
____= (n-1)×5
____= (n-1)
therefore n=____
there are ____two-digit numbers divisible by 5.
The annual profits earned by 30 shops of a shopping complex in a locality give rise to the following distribution:
Draw both ogives for the above data and hence obtain the median.
|
Profit (in lakhs in ₹)
|
No. of shops (frequency)
|
|
More than or equal to 5
|
30
|
|
More than or equal to 10
|
28
|
|
More than or equal to 15
|
16
|
|
More than or equal to 20
|
14
|
|
More than or equal to 25
|
10
|
|
More than or equal to 30
|
7
|
|
More than or equal to 35
|
3
|
Write the coordinates of a point on x-axis which is equidistant from the points $(-3, 4)$ and $(2, 5)$.
→