Question
Prove the following identities:
$1+\frac{\tan^2\theta}{(1+\sec\theta)}=\sec\theta$
$1+\frac{\tan^2\theta}{(1+\sec\theta)}=\sec\theta$
✓
Answer
$\text{L.H.S.}=1+\frac{\tan^2\theta}{(1+\sec\theta)}=\frac{1+\sec\theta+\tan^2\theta}{(1+\sec\theta)}$
$=\frac{\sec^2\theta+\sec\theta}{(1+\sec\theta)}$ $\Big[\because\big(1+\tan^2\theta\big)=\sec^2\theta\Big]$
$=\frac{\sec\theta(1+\sec\theta)}{(1+\sec\theta)}=\sec\theta$
$=\text{R.H.S.}$
Hence, LHS = RHS.
$=\frac{\sec^2\theta+\sec\theta}{(1+\sec\theta)}$ $\Big[\because\big(1+\tan^2\theta\big)=\sec^2\theta\Big]$
$=\frac{\sec\theta(1+\sec\theta)}{(1+\sec\theta)}=\sec\theta$
$=\text{R.H.S.}$
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 triangle $\text{ABC}$ is drawn to circumscribe a circle of radius $4 \ cm$ such that the segments $BD$ and $DC$ into which $BC$ is divided by the point of contact $D$ are of lengths $6 \ cm$ and $8 \ cm$ respectively. Find the lengths of the sides $AB$ and $AC.$
→The shadow of a tower standing on a level ground is found to be 40m longer when Sun's altitude is 30° than when it was 60°. Find the height of the tower.
In an AP: $a _{12} = 37, d = 3$, find a and $S_{12}.$
→The amount of money in the account every year, when ₹ $10000$ is deposited at compound interest at $8\ %$ per annum. Is this situation make an arithmetic progression and why?
→Prove that the points (a, 0), (0, b) and (1, 1) are collinear if $\frac{1}{\text{a}}+\frac{1}{\text{b}}=1.$
→The arithmetic mean of the following data is $14$, find the value of k.
→| $x$ | $5$ | $10$ | $15$ | $20$ | $25$ |
| $f$ | $7$ | $k$ | $8$ | $4$ | $5$ |
Very-short and Short-Answer Questions.
Write the value of $\big(1+\tan^2\theta\big)\cos^2\theta.$
→Write the value of $\big(1+\tan^2\theta\big)\cos^2\theta.$
From each of the two opposite corners of a square of side 8cm, a quadrant of a circle of radius $1.4cm$ is cut. Another circle of radius $4.2cm$ is also cut from the centre as shown in the following figure. Find the area of the remaining (Shaded) portion of the square.$\Big(\text{Use }\pi=\frac{22}{7}\Big)$
→Prove that if x and y are both odd positive integers then $x^2 + y^2$ is even but not divisible by $4$.
→In the given figure, $O$ is the centre of the circle and $TP$ is the tangent to the circle from an external point $T$ . If $\angle P B T=30^{\circ}$, prove that $BA : AT =2: 1$.
→