Prove that: + 3A+ 5A+ 7A=4 2A 4A - Vidyadip

Question

Prove that:
$\cos\text{A}+\cos3\text{A}+\cos5\text{A}+\cos7\text{A}=4\cos\text{A}\cos2\text{A}\cos4\text{A}$

✓

Answer

We have,
$\text{LHS}=\cos\text{A}+\cos3\text{A}+\cos5\text{A}+\cos7\text{A}$
$=\ [\cos3\text{A}+\cos\text{A}]+[\cos7\text{A}+\cos5\text{A}]$
$=\ \Big[2\cos\Big(\frac{3\text{A}+\text{A}}{2}\Big)\cos\Big(\frac{3\text{A}-\text{A}}{2}\Big)\Big]+\Big[2\cos\Big(\frac{7\text{A}+5\text{A}}{2}\Big)\cos\Big(\frac{7\text{A}-5\text{A}}{2}\Big)\Big]$
$=\ 2\cos2\text{A}\cos\text{A}+2\cos6\text{A}\cos4\text{A}$
$=\ 2\cos\text{A}[\cos2\text{A}+\cos6\text{A}]$
$=\ 2\cos\text{A}[\cos2\text{A}+\cos6\text{A}]$
$=\ 2\cos2\text{A}\Big[2\cos\Big(\frac{6\text{A}+2\text{A}}{2}\Big)\cos\Big(\frac{6\text{A}-\text{A}}{2}\Big)\Big]$
$=\ 4\cos\text{A}[\cos4\text{A}\cos2\text{A}]$
$=\ \text{RHS}$
$\therefore\ \cos\text{A}+\cos3\text{A}+\cos5\text{A}+\cos7\text{A}=4\cos\text{A}\cos2\text{A}\cos4\text{A}$
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 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

Solve the following equations:
$4\sin\text{x}\cos\text{x}+2\sin\text{x}+2\cos\text{x}+1=0$

Find the general solution for each of the following equations:
$\sin\text{x}+\sin3\text{x}+\sin5\text{x}=0$

For any two sets A and B, prove the following:
$\text{A}\cap\text{(A}'\cup\text{B})=\text{A}\cap\text{B}$

Find the total number of permutations of the letters of the word 'INSTITUTE'.

If A = {-1, 1}, find A × A × A.

If the coefficients of (2r + 4)th and (r - 2)th teram in the expansion of (1 + x)¹⁸ are equal, find r.

Differentiate the following from first principle

$\sin\sqrt{2\text{x}}$

Find the numberof observation lying between $\overline{\text{X}}-\text{M.D. }$ and $\overline{\text{X}} +\text{ M.D.}$ is the mean deviation from the mean.
38, 70, 48, 34, 63, 42, 55, 44, 53, 47

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow-\infty}\big(\sqrt{\text{x}^2-8\text{x}}+\text{x}\big)$

Find the mean, and standard deviation for the following data:

Marks:	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
Frequency:	1	6	6	8	8	2	2	3	0	2	1	0	0	0	1