Download App

Transformation Formulae — MATHS STD 11 Science — Question

Rajasthan BoardEnglish MediumSTD 11 ScienceMATHSTransformation Formulae5 Marks
Question
Prove that:
$\frac{\sin3\text{A}+\sin5\text{A}+\sin7\text{A}+\sin9\text{A}}{\cos3\text{A}+\cos5\text{A}+\cos7\text{A}\cos9\text{A}}=\tan6\text{A}$
✓

Answer

We have,
$\text{LHS}=\frac{\sin3\text{A}+\sin5\text{A}+\sin7\text{A}+\sin9\text{A}}{\cos3\text{A}+\cos5\text{A}+\cos7\text{A}+\cos9\text{A}}$
$=\ \frac{(\sin9\text{A}+\sin3\text{A})+(\sin7\text{A}+\sin5\text{A})}{(\cos9\text{A}+\cos3\text{A})+(\cos7\text{A}+\cos5\text{A})}$
$=\ \frac{2\sin\Big(\frac{9\text{A}+3\text{A}}{2}\Big)\cos\Big(\frac{9\text{A}-3\text{A}}{2}\Big)+2\sin\Big(\frac{9\text{A}-3\text{A}}{2}\Big)\cos\Big(\frac{7\text{A}-5\text{A}}{2}\Big)}{2\cos\Big(\frac{9\text{A}+3\text{A}}{2}\Big)\cos\Big(\frac{9\text{A}-3\text{A}}{2}\Big)+2\cos\Big(\frac{7\text{A}+5\text{A}}{2}\Big)\cos\Big(\frac{7\text{A}-5\text{A}}{2}\Big)}$
$=\ \frac{2\sin6\text{A}\cos3\text{A}+2\sin6\text{A}\cos\text{A}}{2\cos6\text{A}\cos3\text{A}+2\cos6\text{A}\cos\text{A}}$
$=\ \frac{2\sin6\text{A}(\cos3\text{A}+\cos\text{A})}{2\cos6\text{A}(\cos3\text{A}+\cos\text{A})}$
$=\ \frac{\sin6\text{A}}{\cos6\text{A}}$
$=\ \tan6\text{A}$
$=\ \text{RHS}$
$\therefore\ \frac{\sin3\text{A}+\sin5\text{A}+\sin7\text{A}+\sin9\text{A}}{\cos3\text{A}+\cos5\text{A}+\cos7\text{A}\cos9\text{A}}=\tan6\text{A}$ Hence proved.

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 Transformation Formulae→Transformation Formulae — all question types→MATHS STD 11 Science question papers→Rajasthan Board STD 11 Science question papers→Rajasthan Board question paper generator→

Similar questions

An experiment involves rolling a pair of dice and recording the numbers that come up. Describe the following events:
A: the sum is greater than 8, B: 2 occurs on either die.
C: the sum is at least 7 and a multiple of 3
which pairs of these events are mutually exclusive?
→
Prove that:
$\frac{\sin11\text{A}\sin\text{A}+\sin7\text{A}\sin3\text{A}}{\cos11\text{A}\sin\text{A}+\cos7\text{A}\sin3\text{A}}=\tan8\text{A}$
→
If θ is the angle which the straight line joining the points (x1, y1) and (x2, y2) subtends at the origin, prove that $\tan\theta=\frac{\text{x}_2\text{y}_1-\text{x}_1\text{y}_2}{\text{x}_1\text{x}_2+\text{y}_1\text{y}_2}$ and $\cos\theta=\frac{\text{x}_1\text{x}_2+\text{y}_1\text{y}_2}{\sqrt{\text{x}_1^2+\text{y}_1^2}\sqrt{\text{x}_2^2+\text{y}_2^2}}$
→
The product of three numbers in G.P. is 125 and the sum of their products taken in pairs is $87\frac{1}{2}.$ Find them.
→
prove that:
$\sin(\text{B}-\text{C})\cos(\text{A}-\text{D})+\sin(\text{C}-\text{A})\cos(\text{B}-\text{D})+\sin(\text{A}-\text{B})\cos(\text{C}-\text{D})=0$
→
Find the equation to the straight line which bisects the distance between the points (a, b), (a', b') and also bisects the distance between the points (-a, b) and (a', -b').
→
Prove that:
$\frac{\sin(\theta+\phi)-2\sin\theta+\sin(\theta-\phi)}{\cos(\theta+\phi)-2\cos\theta+\cos(\theta-\phi)}=\tan\theta$
→
$\frac{\text{c}}{\text{a}+\text{b}}=\frac{1-\tan\big(\frac{\text{A}}{2}\big)\tan\big(\frac{\text{B}}{2}\big)}{1+\tan\big(\frac{\text{A}}{2}\big)\tan\big(\frac{\text{B}}{2}\big)}$
→
A man accepts a position with an initial salary of ₹ 5200 per month. It is understood that he will receive an automatic increase of ₹ 320 in the very next month and each month thereafter.
  1. Find his salary for the tenth month.
  2. What is his total earnings during the first year?
→
Find the equation of lines passing through (1, 2) and making angle 30° with y-axis.
→