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 Trigonometric Functions→Trigonometric Functions — all question types→Maths STD 11 Science question papers→CBSE Board STD 11 Science question papers→CBSE Board question paper generator→

Similar questions

A man starts repaying a loan as first instalment of $₹ 100 = 00$. If he increases the instalments by $₹ 5$ every month, what amount he will pay in the $30\ th$ instalment?

If $\theta_1,\ \theta_2,\ \theta_3,\ ...\theta_\text{n}$ are in AP. whose common difference is d, show thet $\sec\theta_1\sec\theta_2+\sec\theta_2\sec\theta_3+...+\sec\theta_{\text{n}-1}\sec\theta_\text{n}=\frac{\theta_\text{n}-\tan\theta_1}{\sin\text{d}}$

Find the angles between the following pairs of straight lines:
3x + 4y - 7 = 0 and 4x - 3y + 5 = 0

Prove that: $\cot x+\cot \left(\frac{\pi}{3}+x\right)+\cot \left(\frac{2 \pi}{3}+x\right)=3 \cot 3 x$

If $\tan\text{x}=\frac{\sin\alpha-\cos\alpha}{\sin\alpha+\cos\alpha},$ then show that $\sin\alpha+\cos\alpha=\sqrt{2}\cos\text{x}.$

One side of equilateral triangle is 18 cm. The mid-points of its sides are joined to from another triangle whose mind-points, in turn, are joined to from still another triangle. the process is continued indefinitely. Find the sum of the (i) Perimeters of all the triangles. (ii) Areas of all triangles.

Prove the following statement by principle of mathematical induction:
$1 + 2 + 2^2 + ...... + 2n = 2^{n+1} - 1$ for all natural numbers.

In how many ways can one select a circket team of eleven from17 players in which only 5 persons can bowl if each cricket team of 11 must include exactly 4 bowlers?

Find the equations to the sides of an isosceles right angled triangle the equation of whose hypotenues is $3x + 4y = 4$ and the opposite vertex is the point $(2, 2).$

Evaluate the following limit:
$\lim\limits_{\text{n}\rightarrow\infty}\frac{\sin\big(\frac{\text{a}}{2^{\text{n}}}\big)}{\sin\big(\frac{\text{b}}{2^{\text{n}}}\big)}$