Question

what would be the value of cos10*cos30*cos50*cos70

Answer 1

Are you allowed to use a calculator?

Answer 2

nope , algebraically solving using trig. identities and formulas

Answer 3

A couple of constants:

\[\cos(60) = \frac{1}{2} \]
\[\cos(30)=\frac{\sqrt{3}}{2}\]

Also, we have:
\[\cos(180 - u) = -\cos(u)\]

And, the Product to Sum Formula:

\[\cos(u) \times \cos(v) = \frac{1}{2}\left[ \cos(u-v) + \cos(u+v) \right]\]

The Problem:

\[ \cos(10) \times \cos(30) \times \cos(50) \times \cos(70) =\]
\[ = \cos(10) \times \frac{\sqrt{3}}{2} \times \cos(50) \times \cos(70) =\]
\[ = \frac{\sqrt{3}}{2} \times \cos(70) \times \cos(10) \times \cos(50) =\]
\[ = \frac{\sqrt{3}}{2} \times \cos(70) \times \frac{1}{2} \times \left[ \cos(40) + \cos(60) \right] =\]
\[ = \frac{\sqrt{3}}{4} \times \cos(70) \times \left[ \cos(40) + \frac{1}{2} \right] =\]
\[ = \frac{\sqrt{3}}{4} \times \left[ \cos(70) \times \cos(40) + \frac{1}{2} \times \cos(70) \right] =\]
\[ = \frac{\sqrt{3}}{4} \times \left[ \frac{1}{2} \times \left[ \cos(30) + \cos(110) \right] + \frac{1}{2} \times \cos(70)\right] =\]
\[ = \frac{\sqrt{3}}{8} \times \left[ \ \cos(30) + \cos(110) + \cos(70) \right] =\]
\[ = \frac{\sqrt{3}}{8} \times \left[ \ \cos(30) - \cos(70) + \cos(70) \right] =\]
\[ = \frac{\sqrt{3}}{8} \times \left[ \ \cos(30) \right] =\]
\[ = \frac{\sqrt{3}}{8} \times \frac{\sqrt{3}}{2} =\]
\[ = \frac{3}{16}\]