Prove that: 5x=5 -20 ^3x+16 ^5x - Vidyadip

Question

Prove that:
$\sin5\text{x}=5\sin\text{x}-20\sin^3\text{x}+16\sin^5\text{x}$

✓

Answer

$\text{LHS}=\sin5\text{x}=\sin(3\text{x}+2\text{x})$
$=\sin3\text{x}\cos2\text{x}+\cos3\text{x}.\sin2\text{x}$
$=(3\sin\text{x}-4\sin^3\text{x})(1-2\sin^3\text{x})\\+(s\cos^3\text{x}=3\cos\text{x})2\sin\text{x}\cos\text{x}.$
$=3\sin\text{x}-4\sin^3\text{x}-6\sin^3\text{x}\\+8\sin^5\text{x}(8\cos^4\text{x}-6\cos^2\text{x})\sin\text{x}$
$=3\sin\text{x}-10\sin^3\text{x}+8\sin^5\text{x}\\+8\sin\text{x}\big((1-\sin^2\text{x})^2-6\sin\text{x}(1-\sin^2\text{x})\big)$
$=13\sin\text{x}-10\sin^3\text{x}+8\sin^5\text{x}+8\sin\text{x}\\-16\sin^3\text{x}8\sin^5\text{x}-6\sin\text{x}+6\sin^3\text{x}$
$=5\sin\text{x}-20\sin^3\text{x}+16\sin^5\text{x}=\text{RHS}$

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

If a circle passes through the point (0, 0),(a, 0),(0, b) then find the coordinates of its centre.

If x^p occurs in the expansion of $\Big(\text{x}^2+\frac{1}{\text{x}}\Big)^{2\text{n}},$ prove that its coefficient is $\frac{2\text{n}!}{\Big(\frac{4\text{n}-\text{p}}{3}\Big)!\Big(\frac{2\text{n}+\text{p}}{3}\Big)!}.$

i. If $f(x)=\left\{\begin{array}{cl}|x|+1, & x<0 \\ 0, & x=0 \\ |x|-1, & x>0\end{array}\right.$, for what values (s) of a does $\lim _{x \rightarrow a} f(x)$ exist?
ii. Find the derivative of the function $\cos \left(x-\frac{\pi}{8}\right)$ from the first principle.

→

If $\cos\text{x}=-\frac{3}{5}$ and x lies in the IIIrd quadrant, find the values of $\cos\frac{\text{x}}{2},\sin\frac{\text{x}}{2},\sin2\text{x}.$

Show that the solution set of the following linear in equations is an unbounded set:
$\text{x}+\text{y}\geq9,3\text{x}+\text{y}\geq12,\text{x}\geq0,\text{y}\geq0.$

Find the equations of the circles passing through two points on y-axis at distances 3 from the origin and having radius 5.

Solve the following equations:
$3-2\cos\text{x}-4\sin\text{x}-\cos2\text{x}+\sin2\text{x}=0$

A bag contains 6 red, 4 white and 8 blue balls. If three balls are drawn at random, find the probability that:
i. one is red and two are white
ii. two are blue and one is red
iii. one is red.

Show that $\text{x}^2+\text{xy}+\text{y}^2,\ \text{z}^2+\text{zx}+\text{x}^2$ and $\text{y}^2+\text{yz}+\text{z}^2$ are consecutive terms of an A.P., if x, y and z are in A.P.

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow1}\frac{(2\text{x}-3)\big(\sqrt{\text{x}}-1\big)}{3\text{x}^2+3\text{x}-6}$