Question

Prove that cos10° cos30° cos50°cos70° = 3\16

Answer 1

please find the attachment

Answer 2

It is prove that$cos 10\textdegree cos30\textdegree cos50\textdegree cos 70\textdegree =\frac{3}{16}$

Step-by-step explanation:

Given:

$cos 10\textdegree cos30\textdegree cos50\textdegree cos 70\textdegree =\frac{3}{16}$

To Find:

To prove that $cos 10\textdegree cos30\textdegree cos50\textdegree cos 70\textdegree =\frac{3}{16}$

Formula Used:

$2 cos P cos Q = cos ( P+Q) + cos(P-Q)$  -------------------------- Identity no.01

Solution:

As given-$cos 10\textdegree cos30\textdegree cos50\textdegree cos 70\textdegree =\frac{3}{16}$

LHS   $=cos 10\textdegree cos30\textdegree cos50\textdegree cos 70\textdegree$

        $= cos 30\textdegree( cos10\textdegree cos50\textdegree) cos 70\textdegree$

        $= \frac{\sqrt{3} }{2}.\frac{1}{2} ( 2cos10\textdegree cos50\textdegree) cos 70\textdegree$

Applying identity no. 01.  $2 cos P cos Q = cos ( P+Q) + cos(P-Q)$

         $= \frac{\sqrt{3} }{2}.\frac{1}{2} ( cos60\textdegree +cos40\textdegree) cos 70\textdegree$

         $= \frac{\sqrt{3} }{4} ( cos60\textdegree cos70\textdegree +cos40\textdegree cos70\textdegree)$

Using value of $cos60\textdegree=\frac{1}{2}$

        $= \frac{\sqrt{3} }{4} ( \frac{1}{2} cos70\textdegree +cos40\textdegree cos70\textdegree)$

        $= \frac{\sqrt{3} }{8} ( \ cos70\textdegree + 2cos40\textdegree cos70\textdegree)$

Applying identity no. 01.  $2 cos P cos Q = cos ( P+Q) + cos(P-Q)$

        $= \frac{\sqrt{3} }{8} ( \ cos70\textdegree + cos110\textdegree + cos30\textdegree)$

        $= \frac{\sqrt{3} }{8} ( \ cos70\textdegree + cos(180\textdegree- 70\textdegree) + cos30\textdegree)$

Using value of $cos(180\textdegree-70\textdegree) = - cos 70 \textdegree$  and $cos30\textdegree=\frac{\sqrt{3} }{2}$

        $= \frac{\sqrt{3} }{8} ( \ cos70\textdegree + cos- 70\textdegree) + \frac{\sqrt{3} }{2} )$

       $= \frac{\sqrt{3} }{8} ( \frac{\sqrt{3} }{2} )$

       $=\frac{3}{16}$

      = RHS

Because LHS= RHS

Hence ,  it is proved that $cos 10\textdegree cos30\textdegree cos50\textdegree cos 70\textdegree =\frac{3}{16}$