Download App

Trigonometric Functions — MATHS STD 11 Science — Question

Rajasthan BoardEnglish MediumSTD 11 ScienceMATHSTrigonometric Functions5 Marks
Question
Prove that:
$\cos\text{x}\cos\frac{\text{x}}{2}-\cos3\text{x}\cos\frac{9\text{x}}{2}=\sin7\text{x}\sin8\text{x}.$
✓

Answer

Consider the left hand side of the given expression:
$\text{LHS}=\cos\text{x}\cos\frac{\text{x}}{2}-\cos{3\text{x}}\cos\frac{9\text{x}}{2}$
We know that $2\cos\text{A}\cos\text{B}=\cos\text{A+B}+\cos(\text{A}-\text{B})$
Thus,
$\text{LHS}=\frac{1}{2}\Big[\cos\Big(\text{x}+\frac{\text{x}}{2}\Big)+\cos\Big(\text{x}-\frac{\text{x}}{2}\Big)-\frac{1}{2}\Big[\cos\Big(3\text{x}+\frac{9\text{x}}{2}\Big)+\cos\Big(3\text{x}-\frac{9\text{x}}{2}\Big)\Big]$
$=\ \frac{1}{2}\Big[\cos\Big(\frac{3\text{x}}{2}\Big)+\cos\Big(\frac{\text{x}}{2}\Big)\Big]-\frac{1}{2}\Big[\cos\Big(\frac{15\text{x}}{2}\Big)+\cos\Big(-\frac{3\text{x}}{2}\Big)\Big]$
$=\ \frac{1}{2}\Big[\cos\Big(\frac{3\text{x}}{2}\Big)-\cos\Big(\frac{\text{x}}{2}\Big)\Big]-\frac{1}{2}\Big[\cos\Big(\frac{15\text{x}}{2}\Big)+\cos\Big(-\frac{3\text{x}}{2}\Big)\Big]$$[\because\ \cos(-\theta)=\cos\theta]$
$=\ \frac{1}{2}\Big[\cos\Big(\frac{3\text{x}}{2}\Big)+\cos\Big(\frac{\text{x}}{2}\Big)-\cos\Big(\frac{15\text{x}}{2}\Big)-\cos\Big(\frac{3\text{x}}{2}\Big)\Big]$
$=\ \frac{1}{2}\Big[\cos\Big(\frac{\text{x}}{2}\Big)-\cos\Big(\frac{15\text{x}}{2}\Big)\Big]$
Also we know that,
$\cos\text{D}-\cos\text{C}=2\sin\frac{\text{C+D}}{2}\sin\frac{\text{C}-\text{D}}{2}$
Therefore,
$\text{LHS}=\frac{1}{2}\times2\sin\frac{\frac{15\text{x}}{2}+\frac{\text{x}}{2}}{2}\sin\frac{\frac{15\text{x}}{2}-\frac{\text{x}}{2}}{2}$
$=\ \sin\frac{\frac{16\text{x}}{2}}{2}\sin\frac{\frac{14\text{x}}{2}}{2}$
$=\ \sin\frac{8\text{x}}{2}\sin\frac{7\text{x}}{2}$
$=\ \text{RHS}$
Note: Question given in the book is incorrect.
RHS should be equal to $\sin\frac{8\text{x}}{2}\sin\frac{7\text{x}}{2}.$

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 Functions→Trigonometric Functions — all question types→MATHS STD 11 Science question papers→Rajasthan Board STD 11 Science question papers→Rajasthan Board question paper generator→

Similar questions

 Prove the following:
$\sin(\text{n}+1)\text{x}\sin(\text{n}+2)\text{x}+2\cos(\text{n}+1)\text{x}\cos(\text{n}+2)\text{x}=\cos\text{x}$
→
Differentiate the following function with respect to x:

$\log_{\text{x}^2}\text{x}$

 

→
Prove that:
$\frac{\cos3\text{A}+2\cos5\text{A}+\cos7\text{A}}{\cos\text{A}+2\cos3\text{A}+\cos5\text{A}}=\frac{\cos5\text{A}}{\cos3\text{A}}$
→
If A = {1, 2, 3}, B = {3, 4} and C = {4, 5, 6}, find
$(\text{A}\times\text{B})\cap(\text{A}\times\text{C})$
→
Find the equations to the straight lines which pass through the origin and are inclined at an angle of 75° to the straight line $\text{x}+\text{y}+\sqrt{3}(\text{y}-\text{x})=\text{a}.$
→
If a, b, c, are in G.P., prove that the following are also in G.P.
$\text{a}^2+\text{b}^2,\text{ab}+\text{bc},\text{b}^2+\text{c}^2$
→
If $\tan\text{x}=\frac{\text{b}}{\text{a}},$ then find the value of $\sqrt{\frac{\text{a}+\text{b}}{\text{a}-\text{b}}}+\sqrt{\frac{\text{a}-\text{b}}{\text{a}+\text{b}}}$
→
Find the mean deviation from the mean for following data:
xi
5
10
15
20
25
fi
7
4
6
3
5
→
Find the general solution for each of the following equations:
$\sin2\text{x}+\cos\text{x}=0$
→

Prove that the term independent of x in the expansion of $\Big(\text{x}+\frac{1}{\text{x}}\Big)^{2\text{n}}$ is $\frac{1,3,5.....(2\text{n}-1)}{\text{n}!}.2^{\text{n}}.$

→