Prove that: 3A+ 5A+ 7A+ 15A=4 4A 5A 6A - Vidyadip

Question

Prove that:
$\cos3\text{A}+\cos5\text{A}+\cos7\text{A}+\cos15\text{A}=4\cos4\text{A}\cos5\text{A}\cos6\text{A}$

✓

Answer

We have,
$\text{LHS}=\cos3\text{A}+\cos5\text{A}+\cos7\text{A}+\cos15\text{A}$
$=\ [\cos5\text{A}+\cos3\text{A}]+[\cos15\text{A}+\cos7\text{A}]$
$=\ \Big[2\cos\frac{5\text{A}+3\text{A}}{2}\cos\frac{5\text{A}-3\text{A}}{2}\Big]+\Big[2\cos\frac{15\text{A}+7\text{A}}{2}\cos\frac{15\text{A}-7\text{A}}{2}\Big]$
$=\ 2\cos4\text{A}\cos\text{A}+2\cos11\text{A}\cos4\text{A}$
$=\ 2\cos4\text{A}[\cos\text{A}+\cos11\text{A}]$
$=\ 2\cos4\text{A}[\cos11\text{A}+\cos\text{A}]$
$=\ 2\cos4\text{A}\Big[2\cos\frac{(11\text{A}+\text{A)}}{2}\cos\frac{(11\text{A}-\text{A})}{2}\Big]$
$=\ 4\cos\text{A}[\cos6\text{A}\cos5\text{A}]$
$=\ 4\cos4\text{A}\cos5\text{A}\cos6\text{A}$
$=\ \text{RHS}$
$\therefore\ \cos3\text{A}+\cos5\text{A}+\cos7\text{A}+\cos15\text{A}=4\cos4\text{A}\cos5\text{A}\cos6\text{A}$
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

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow{\frac{\pi}{3}}}\frac{\sqrt{3}-\tan\text{x}}{\pi-3\text{x}}$

Find the equation of the circle whose diameter is the line segment joining (-4, 3) and (12, -1). Find also the intercept made by it on y-axis.

The mean and variance of 7 observations are 8 and 16 respectively. If five of the observations are 2, 4, 10, 12, 14 find the remaining two observations.

Prove the following by the principle of mathematical induction:
$1^2 + 2^2 + 3^2 +...+ \text{n}^2=\frac{\text{n}(\text{n}+1)(2\text{n}+1)}{6}$

The perpendicular distance of a line from the origin is 5 units and its slope is -1. Find the equation of the line.

Calculate the mean deviation about the median of the following observation:
22, 24, 30, 27, 29, 31, 25, 28, 41, 42

$\sin(\text{B}-\text{C})\cos(\text{A}-\text{D})+\sin(\text{C}-\text{A})\\\cos(\text{B}-\text{D})+\sin(\text{A}-\text{B})\cos(\text{C}-\text{D})=0$

$\text{a}\sin\frac{\text{A}}{2}\sin\Big(\frac{\text{B}-\text{C}}{2}\Big)+\text{b}\sin\frac{\text{B}}{2}\sin\Big(\frac{\text{C}-\text{A}}{2}\Big)+\text{c}\sin\frac{\text{C}}{2}\sin\Big(\frac{\text{A}-\text{B}}{2}\Big)=0.$

Find the sum of the following geometric series:

$(\text{x}+\text{y})+(\text{x}^2+\text{xy}+\text{y}^2)+(\text{x}^3+\text{x}^2\text{y}+\text{xy}^2+\text{y})+\ ...\text{ to n terms;}$

The mean and standard deviation of six observation are 8 and 4 respectively. If each observation is multiplied by 3, find the new mean and new standard deviation of the resulting observations.