Question

Show that cos40°cos100°cos160°=1/8

Answer 1

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Answer 2

To Prove: cos40°cos100°cos160° = 1/8.

          $L.H.S= cos40^0cos100^0cos160^0=\frac{1}{2}[2cos160^0cos40^0]cos100^0$

Using the formula: 2 cosA cosB = cos(A+B) + cos(A-B)

                                → cos (180° - α) = - cos α

                  $L.H.S.=\frac{1}{2} [cos200^0+cos120^0]cos100^0$

                              $=\frac{1}{2} [cos(180^0+20^0)+cos(180^0-60^0)]cos100^0\\\\=\frac{1}{2}[-cos20^0-cos60^0]cos100^ 0\\\\=\frac{1}{2}[-cos20^0cos100^0-\frac{1}{2}cos100^0]\\\\= \frac{1}{2}[-cos20^0cos100^0]-\frac{1}{4}cos100^0\\\\ =\frac{-1}{4}[2cos100^0cos20^0]-\frac{1}{4}cos100^0 \\\\=\frac{-1}{4}[cos(100^0+20^0)+cos(100^0-20^0)]-\frac{1}{4}cos100^0 \\\\=\frac{-1}{4}[cos120^0+cos80^0]-\frac{1}{4}cos100^0$

                              $=\frac{-1}{4}[cos(180^0-60^0)+cos(180^0-100^0)]-\frac{1}{4}cos100^0\\\\=\frac{-1}{4}[-cos60^0-cos100^0]-\frac{1}{4}cos100^0\\\\ =\frac{1}{4}cos60^0+\frac{1}{4}cos100^0 -\frac{1}{4}cos100^0\\\\=\frac{1}{4}cos60^0 \\\\=\frac{1}{8}=R.H.S.$

                          L.H.S. = R.H.S. Proved

               ∴ cos40° cos100° cos 160° = 1/8 Proved

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