Download App

Trigonometric Identities — Maths STD 10 — Question

CBSE BoardEnglish MediumSTD 10MathsTrigonometric Identities5 Marks
Question
If $\tan\text{A}=\text{n}\tan\text{B}$ and $\sin\text{A}=\text{m}\sin\text{B},$ prove that $\cos^2\text{A}=\frac{\big(\text{m}^2-1\big)}{\text{n}^2-1}.$
✓

Answer

We have, $\tan\text{A}=\text{n}\tan\text{B}$
$\Rightarrow\cot\text{B}=\frac{\text{n}}{\tan\text{A}}\dots(\text{i})$
Again, $\sin\text{A}=\text{m}\sin\text{B}$
$\Rightarrow\text{cosec B}=\frac{\text{m}}{\sin\text{A}}\dots(\text{ii})$
Squaring (i) and (ii) and subtracting (ii) from (i), we get:
$\frac{\text{m}^2}{\sin^2\text{A}}-\frac{\text{n}^2}{\tan^2\text{A}}=\text{cosec}^2\text{B}-\cot^2\text{B}$
$\Rightarrow\frac{\text{m}^2}{\sin^2\text{A}}-\frac{\text{n}^2\cos}{\sin^2\text{A}}=1$
$\Rightarrow\text{m}^2-\text{n}^2\cos^2\text{A}=\sin^2\text{A}$
$\Rightarrow\text{m}^2-\text{n}^2\cos^2\text{A}=1-\cos^2\text{A}$
$\Rightarrow\text{n}^2\cos^2\text{A}-\cos^2\text{A}=\text{m}^2-1$
$\Rightarrow\cos^2\text{A}\big(\text{n}^2-1\big)=\big(\text{m}^2-1\big)$
$\Rightarrow\cos^2\text{A}=\frac{\big(\text{m}^2-1\big)}{\big(\text{n}^2-1\big)}$
$\therefore\ \cos^2\text{A}=\frac{\big(\text{m}^2-1\big)}{\big(\text{n}^2-1\big)}$

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

Similar questions

A solid toy is in the form of a hemisphere surmounted by a right circular cone. The height of cone is $4 \ cm$ and the diameter of the base is $8 \ cm.$ Determine the volume of the toy. If a cube circumscribes the toy, then find the difference of the volumes of cube and the toy. Also, find the total surface area of the toy.
→
A train, travelling at a uniform speed for 360km, would have taken 48 minutes less to travel the same distance if its speed were 5km/hr more. Find the original speed of the train.
→
The sum of first 9 terms of an A.P. is 162 . The ratio of its $6^{\text {th }}$ term to its $13^{\text {th }}$ term is $1: 2$. Find the first and $15^{\text {th }}$ term of the A.P?
→
50 students enter for a school javelin throw competition. The distance (in metres) thrown are recorded below:
Distance (in m)
0-20
20-40
40-60
60-80
80-100
Number of students
6
11
17
12
4
  1. Construct a cumulative frequency table.
  2. Draw a cumulative frequency curve (less than type) and calculate the median distance thrown by using this curve.
  3. Calculate the median distance by using the formula for median.
  4. Are the median distance calculated in (ii) and (iii) same?
→
The sum of the digits of a two-digit number is 15. The number obtained by interchanging the digits exceeds the given number by 9. Find the number.
→
A contract on constrution job specifies a penalty for delay of completion beyond a certain date as follows: ₹ 200 for the first day, ₹ 250 for the second day, ₹ 300 for the third day, etc, the penalty for each succeeding day being ₹ 50 more than for the preceding day. How much money the contractor has to pay as penalty, if he has delayed the work by 30 days?
→
Sum of the first 14 terms of an AP is 1505 and its first term is 10. Find its $25^{th}$​​​​​​​ term.
→
Draw a circle of radius 3cm. Take two points P and Q on one of its extended diameter each at a distance of 7cm from its centre. Draw tangents to the circle from these two points P and Q.
→
Evaluate the following:
If A = 60° and B = 30°, verify that:
$\tan(\text{A}-\text{B})=\frac{\tan\text{A}-\tan\text{B}}{1+\tan\text{A}\tan\text{B}}$
→
Verity that 3, -2, 1 are the zeros of the cubic polynomial $p(x) = x^3 - 2x^2 - 5x + 6$ and verify the relation between its zeros and coefficients.
→