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 "Solve the Following Question.(4 Marks)" from Trigonometric Ratios Of Compound Angles→Trigonometric Ratios Of Compound Angles — all question types→Maths STD 11 Science question papers→Maharashtra Board STD 11 Science question papers→Maharashtra Board question paper generator→

Similar questions

$\Bigg|\sin\text{x}\sin\Big(\frac{\pi}{3}-\text{x}\Big)\sin\Big(\frac{\pi}{3}+\text{x}\Big)\Bigg|\not<\frac{1}{4}$ for all values of x.

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow\frac{3\pi}{2}}\frac{1+\text{cosec}^3\text{x}}{\cot^2\text{x}}$

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

Find n and r if,

${ }^n C_{r-1}:{ }^n C_r:{ }^n C_{r+1}=20: 35: 42$

Prove by the method of induction, for all n ∈ N.

$\frac{1}{1.3}+\frac{1}{3.5}+\frac{1}{5.7}+\ldots+\frac{1}{(2 n-1)(2 n+1)}=\frac{n}{2 n+1}$

A railway train is travelling on a circular curve of 1500 metres radius at the rate of 66km/ hr. Through what angle has it turned in 10 seconds?

In each of the following find the equations of the hyperbola satisfying the given conditions:Vertices $(\pm2, 0)$, foci $(\pm3, 0)$ [NCERT]

Find the equation of the circle, if the equations of two diameters are $2x + y = 6$ and $3x + 2y = 4$ and radius is $9.$

If the line $y = mx + 1$ is tangent to the parabola $y^2 = 4x,$ then find the value of $m.$

Find the number of words formed by permuting all the letters of the following words:
CONSTANTINOPLE.