Question
If $\sec A+\tan A=p$, show that: $\sin A =\frac{p^2-1}{p^2+1}$
✓
Answer
$\frac{p^2-1}{p^2+1}$
$=\frac{(\sec A +\tan A )^2-1}{(\sec A +\tan A )^2+1}$
$=\frac{\sec ^2 A +\tan ^2 A +2 \tan A \sec A -1}{\sec ^2 A +\tan ^2 A +2 \tan A \sec A +1}$
$=\frac{\tan ^2 A +\tan ^2 A +2 \tan A \sec A }{\sec ^2 A +\sec ^2 A +2 \tan A \sec A }$
$=\frac{2 \tan ^2 A +2 \tan A \sec A }{2 \sec ^2 A +2 \tan A \sec A }$
$=\frac{2 \tan A (\tan A +\sec A )}{2 \sec A (\tan A +\sec A )}$
$=\sin A $
$=\frac{(\sec A +\tan A )^2-1}{(\sec A +\tan A )^2+1}$
$=\frac{\sec ^2 A +\tan ^2 A +2 \tan A \sec A -1}{\sec ^2 A +\tan ^2 A +2 \tan A \sec A +1}$
$=\frac{\tan ^2 A +\tan ^2 A +2 \tan A \sec A }{\sec ^2 A +\sec ^2 A +2 \tan A \sec A }$
$=\frac{2 \tan ^2 A +2 \tan A \sec A }{2 \sec ^2 A +2 \tan A \sec A }$
$=\frac{2 \tan A (\tan A +\sec A )}{2 \sec A (\tan A +\sec A )}$
$=\sin A $
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
Prove.$\frac{1+\cos A}{1-\cos A}=\frac{\tan ^2 A}{(\sec A-1)^2}$
→In an auditorium, seats were arranged in rows and columns. The number of rows was equal to the number of seats in each row. When the number of rows was doubled and the number of seats in each row was reduced by $10,$ the total number of seats increased by $300.$ Find:
(1) the number of rows in the original arrangement.
(2) the number of seats in the auditorium after re-arrangement.
→(1) the number of rows in the original arrangement.
(2) the number of seats in the auditorium after re-arrangement.
From a solid right circular cylinder with height 10 cm and radius of the base 6 cm, a right circularcone of the same height and same base is removed. Find the volume of the remaining solid.
In the figure, O is the centre of the circle, ∠AOE = 150°, ∠DAO = 51°. Calculate the sizes of the angles CEB and OCE.
Find the 10th term from the end of the A.P 4, 9, 14, ..... 254
The percentage of marks obtained by a student in monthly unit tests are given below:
Based on this data find the probability that the student gets
(iii) less than 65% marks in a unit test.
| Unit Test | l | ll | lll | lV | V | Vl |
| Percentage of marks obtained | 72 | 67 | 69 | 74 | 71 | 76 |
(iii) less than 65% marks in a unit test.
For each inequality, determine which of the given numbers are in the solution set:
16 - 5 x ≤ - 4; 4, -3, 10.
16 - 5 x ≤ - 4; 4, -3, 10.
Use graph paper to answer this question:
(i) Plot the points A (4,6) and B (1, 2).
(ii) A’ is the image of A when reflected in X-axis,
(iii) B’ is the image of B when B is reflected in the line AA’.
(iv) Give the geometrical name for the figure ABA’B’.
(i) Plot the points A (4,6) and B (1, 2).
(ii) A’ is the image of A when reflected in X-axis,
(iii) B’ is the image of B when B is reflected in the line AA’.
(iv) Give the geometrical name for the figure ABA’B’.
A cylinderical container with diameter of base $42\ cm$ contains sufficient water to submerge a rectangular soild of iron with dimesions $22\ cm \times 14\ cm \times 10.5\ cm.$ find the rise in level of the water the solid is submerged.
→Prove the following trigonometric identities.
$\frac{1}{\sec A-1}+\frac{1}{\sec A+1}=2 \operatorname{cosec} A \cot A$
→$\frac{1}{\sec A-1}+\frac{1}{\sec A+1}=2 \operatorname{cosec} A \cot A$