Prove that: 6 42^ 66^ 78^ =1 16 - Vidyadip

Question

Prove that: $\sin6\circ\sin42^\circ\sin66^\circ\sin78^\circ=\frac{1}{16}$

✓

Answer

$\text{LHS}=\cos6^\circ\cos42^\circ\cos66^\circ\cos78^\circ$$=\frac{1}{4}(2\cos6^\circ\cos66^\circ)(2\cos42^\circ\cos78^\circ)$
$=\frac{1}{4}(\cos72^\circ+\cos60^\circ)(\cos120^\circ+\cos36^\circ)$
$=\frac{1}{4}\Big(\sin18^\circ+\frac{1}{2}\Big)\bigg(-\frac{2}{2}+\frac{\sqrt{5}+1}{4}\bigg)$
$=\frac{1}{4}\bigg(\frac{\sqrt{5}-1}{4}+\frac{1}{2}\bigg)\bigg(\frac{\sqrt{5}+1}{4}-\frac{1}{2}\bigg)$
$=\frac{1}{4}\bigg(\frac{\sqrt{5}-1+2}{4}\bigg)\bigg(\frac{\sqrt{5}+1-2}{4}\bigg)$
$=\frac{1}{64}(\sqrt{5}+1)(\sqrt{5}-1)$
$=\frac{1}{64}(\sqrt{5})^2-1^2)$
$=\frac{1}{64}(5-1)$
$=\frac{1}{16}$
$=\text{RHS}$

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 Trigonometric Ratios Of Compounds→Trigonometric Ratios Of Compounds — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Find the derivative of the following functions: $5\sin\text{x} – 6\cos\text{x} + 7$

In a group of 950 person, 750 can speak Hindi and 460 can speak English. Find: how many can speak Hindi only.

Find the derivative of the following functions from first principle: $\cos(\text{x}-\frac{\pi}{8})$

$\text{If}\ \frac{\cos(\text{A}-\text{B})}{\cos(\text{A+B})}+\frac{\cos(\text{C+D})}{\cos(\text{C}-\text{D})}=0,$ prove that $\tan\text{A}\tan\text{B}\tan\text{C}\tan\text{D}=-1$

Find the eccentricity, coordinates of foci, length of the latus-rectum of the following ellipse:$25\text{x}^2+16\text{y}^2=1600.$

Find sum of these numbers in G.P. is $21$ and the sum of their aquares is $189$. Find the numbers.

For any two sets A and B, prove that $\text{A}\cup(\text{B}-\text{A})=\text{A}\cup\text{B}$

If $z_1, z_2$ are two complex numbers such that $|\text{z}_1|=|\text{z}_2|$ and $\text{arg(z}_1)+\text{arg(z}_2)=\pi,$ then show that $\text{z}_1=-\bar{\text{z}}_2.$

Find the equation of the parabola whose: Focus is (1, 1) and the directrix is x + y + 1 = 0

Solve the system of inequality graphically: 2x + y $\geq$ 4, x + y $\le$ 3, 2x – 3y $\le$ 6