Prove that: 40^ 80^ 160=-1 8 - Vidyadip

Question

Prove that: $\cos40^\circ\cos80^\circ\cos160=-\frac{1}{8}$

✓

Answer

$\cos40^\circ\cos80^\circ\cos160^\circ=-\frac{1}{8}$ $\text{LHS}=\cos40^\circ\cos80^\circ\cos160^\circ$ $=\ \cos80^\circ\cos40^\circ\cos160^\circ$ Multiplying and dividing by 2 $=\ \frac{1}{2}(\cos80^\circ\times(2\cos40^\circ\cos160^\circ))$ $2\cos\text{A}\cos\text{B}=\cos(\text{A+B})+\cos(\text{A}-\text{B})$ $=\ \frac{1}{2}(\cos80^\circ(\cos(40^\circ+160^\circ)+\cos(40^\circ-160^\circ)))$ $=\ \frac{1}{2}(\cos80^\circ(\cos200+\cos(-120)))$ $=\ \frac{1}{2}(\cos80^\circ(\cos180^\circ+20^\circ)+\cos(180^\circ-60^\circ))$ $=\ \frac{1}{2}\cos80^\circ(\cos20^\circ+\cos60^\circ)$ $=\ \frac{1}{2}\cos80^\circ\cos20^\circ+\frac{1}{2}\cos80^\circ+60^\circ$ $=\ -\frac{1}{2}(2\cos80^\circ\cos20^\circ)+\frac{1}{2}\cos80^\circ\cos60^\circ$ $=\ -\frac{1}{4}[2\cos80^\circ\cos20^\circ+\cos80^\circ]$ $=\ -\frac{1}{4}[\cos(80^\circ+20^\circ)+\cos(80^\circ-20^\circ)+\cos80^\circ]$ $=\ -\frac{1}{4}[\cos100^\circ+\cos60^\circ+\cos80^\circ]$ $=\ -\frac{1}{4}[\cos(180^\circ-80^\circ)+\cos60^\circ+\cos80^\circ]$ $=\ -\frac{1}{4}[-\cos80^\circ+\cos60^\circ+\cos80^\circ]$ $=\ -\frac{1}{4}\cos60^\circ$ $=\ -\frac{1}{4}\times\frac{1}{2}$ $=\ -\frac{1}{8}\ \text{RHS}$

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 "(Each question 4 marks)" from Transformation Formulae→Transformation Formulae — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Differentiate the following from first principle$\tan^2\text{x} $

Let f and g be two real functions defined by $\text{f(x)}=\sqrt{\text{x}+1}$ and $\text{g(x)}=\sqrt{9-\text{x}^2}$ Then describe the following functions: $\frac{\text{g}}{\text{f}}$

$\text{If}\ \cos(\text{A+B})\sin(\text{C}-\text{D})=\cos(\text{A}-\text{B})\sin(\text{C+D}),$ prove that $\tan\text{A}\tan\text{B}\tan\text{C}+\tan\text{D}=0$

Find the angles between the following pairs of straight lines: 3x + y + 12 = 0 and x + 2y - 1 = 0

Prove the following by the principle of mathematical induction: $1 + 2 + 3 + ... + \text{n}=\frac{\text{n}(\text{n}+1)}{2}$ i.e, the sum of the first n natural numbers is $\frac{\text{n}(\text{n}+1)}{2}.$

Calculate mean deviation about median age distribution of $100$ persons given below:

Age	16-20	21-25	26-30	31-35	36-40	41-45	46-50	51-55
No. of persons	5	6	12	14	26	12	16	9

In each of the followin find the equation of the hyperbola satisying the given conditions: foci $(\pm5, 0)$, transverse axis = 8 [NCERT]

Find three numbers in G.P. whose sum is 38 and their product is 1728.

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

Find the mean and standard deviation using short cut method.

$x_i$	60	61	62	63	64	65	66	67	68
$f_i$	2	1	12	29	25	12	10	4	5