Product Multiple

Geometry Level 2

( 1 + cos ⁡ A ) ( 1 + cos ⁡ 3 A ) ( 1 + cos ⁡ 5 A ) ( 1 + cos ⁡ 7 A ) \large (1+\cos A)(1+\cos 3A)(1+ \cos 5A)(1+\cos 7A)

What is the value of the expression above if A = 22. 5 ∘ A = 22.5^\circ ?

1 4 \frac14 1 8 \frac18 1 2 \frac12 1 3 \frac13

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Avn Bha by avn bha , 22, India

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3 solutions

Chew-Seong Cheong
May 22, 2015

( 1 + cos ⁡ 22. 5 ∘ ) ( 1 + cos ⁡ 67. 5 ∘ ) ( 1 + cos ⁡ 112. 5 ∘ ) ( 1 + cos ⁡ 157. 5 ∘ ) = ( 1 + cos ⁡ 22. 5 ∘ ) ( 1 + cos ⁡ 67. 5 ∘ ) ( 1 − cos ⁡ ( 18 0 ∘ − 112. 5 ∘ ) ) ( 1 − cos ⁡ ( 18 0 ∘ − 157. 5 ∘ ) ) = ( 1 + cos ⁡ 22. 5 ∘ ) ( 1 + cos ⁡ 67. 5 ∘ ) ( 1 − cos ⁡ 22. 5 ∘ ) ( 1 − cos ⁡ 67. 5 ∘ ) = ( 1 − cos ⁡ 2 22. 5 ∘ ) ( 1 − cos ⁡ 2 67. 5 ∘ ) = ( 1 2 − 1 2 ( 2 cos ⁡ 2 22. 5 ∘ − 1 ) ) ( 1 2 − 1 2 ( 2 cos ⁡ 2 67. 5 ∘ − 1 ) ) = 1 4 ( 1 − cos ⁡ 4 5 ∘ ) ( 1 − cos ⁡ 13 5 ∘ ) = 1 4 ( 1 − cos ⁡ 4 5 ∘ ) ( 1 + cos ⁡ 4 5 ∘ ) = 1 4 ( 1 − 1 2 ) ( 1 + 1 2 ) = 1 4 ( 1 − 1 2 ) = 1 8 (1+\cos{22.5^\circ}) (1+\cos{67.5^\circ}) (1+\cos{112.5^\circ}) (1+\cos{157.5^\circ}) \\ = (1+\cos{22.5^\circ}) (1+\cos{67.5^\circ}) \\ \quad \quad (1-\cos{(180^\circ-112.5^\circ)}) (1-\cos{(180^\circ-157.5^\circ)}) \\ = (1+\cos{22.5^\circ}) (1+\cos{67.5^\circ}) (1 - \cos{22.5^\circ}) (1 -\cos{67.5^\circ}) \\ = (1-\cos^2{22.5^\circ}) (1-\cos^2{67.5^\circ}) \\ = \left( \frac{1}{2} - \frac{1}{2} (2\cos^2{22.5^\circ} - 1)\right) \left( \frac{1}{2} - \frac{1}{2} (2\cos^2{67.5^\circ} - 1)\right) \\ = \frac {1}{4} (1 - \cos{45^\circ}) (1 - \cos{135^\circ}) \\ = \frac {1}{4} (1 - \cos{45^\circ}) (1 + \cos{45^\circ}) \\ = \frac {1}{4} (1 - \frac{1}{\sqrt{2}}) (1 + \frac{1}{\sqrt{2}}) \\ = \frac {1}{4} (1 - \frac{1}{2}) \\ = \boxed{\frac{1}{8}}

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Aquilino Madeira
Jul 12, 2015

[ 1 + cos ⁡ ( 22.5 ) ] [ 1 + cos ⁡ ( 67.5 ) ] [ 1 + cos ⁡ ( 112.5 ) ] [ 1 + cos ⁡ ( 157.5 ) ] = [ 1 + cos ⁡ ( 22.5 ) ] [ 1 + cos ⁡ ( 67.5 ) ] [ 1 + cos ⁡ ( 90 + 22.5 ) ] [ 1 + cos ⁡ ( 90 + 67.5 ) ] = [ 1 + cos ⁡ ( 22.5 ) ] [ 1 + cos ⁡ ( 67.5 ) ] [ 1 − sin ⁡ ( 22.5 ) ] [ 1 + sin ⁡ ( 67.5 ) ] = [ 1 + cos ⁡ ( 22.5 ) ] [ 1 + cos ⁡ ( 90 − 22.5 ) ] [ 1 − sin ⁡ ( 22.5 ) ] [ 1 + sin ⁡ ( 90 − 22.5 ) ] = [ 1 + cos ⁡ ( 22.5 ) ] [ 1 + sin ⁡ ( 22.5 ) ] [ 1 − sin ⁡ ( 22.5 ) ] [ 1 − cos ⁡ ( 22.5 ) ] = [ 1 + cos ⁡ ( 22.5 ) ] [ 1 − cos ⁡ ( 22.5 ) ] [ 1 + sin ⁡ ( 22.5 ) ] [ 1 − sin ⁡ ( 22.5 ) ] = [ 1 − cos ⁡ 2 ( 22.5 ) ] [ 1 − sin ⁡ 2 ( 22.5 ) ] = cos ⁡ 2 ( 22.5 ) s i n 2 ( 22.5 ) = cos ⁡ ( 22.5 ) s i n ( 22.5 ) × cos ⁡ ( 22.5 ) s i n ( 22.5 ) = 2 × cos ⁡ ( 22.5 ) s i n ( 22.5 ) 2 × 2 × cos ⁡ ( 22.5 ) s i n ( 22.5 ) 2 = s i n ( 2 × 22.5 ) 2 × s i n ( 2 × 22.5 ) 2 = s i n ( 45 ) 2 × s i n ( 45 ) 2 = 2 2 2 × 2 2 2 = 2 4 4 = 1 8 \begin{array}{l} \left[ {1 + \cos (22.5)} \right]\left[ {1 + \cos (67.5)} \right]\left[ {1 + \cos (112.5)} \right]\left[ {1 + \cos (157.5)} \right]\\ = \left[ {1 + \cos (22.5)} \right]\left[ {1 + \cos (67.5)} \right]\left[ {1 + \cos (90 + 22.5)} \right]\left[ {1 + \cos (90 + 67.5)} \right]\\ = \left[ {1 + \cos (22.5)} \right]\left[ {1 + \cos (67.5)} \right]\left[ {1 - \sin (22.5)} \right]\left[ {1 + \sin (67.5)} \right]\\ = \left[ {1 + \cos (22.5)} \right]\left[ {1 + \cos (90 - 22.5)} \right]\left[ {1 - \sin (22.5)} \right]\left[ {1 + \sin (90 - 22.5)} \right]\\ = \left[ {1 + \cos (22.5)} \right]\left[ {1 + \sin (22.5)} \right]\left[ {1 - \sin (22.5)} \right]\left[ {1 - \cos (22.5)} \right]\\ = \left[ {1 + \cos (22.5)} \right]\left[ {1 - \cos (22.5)} \right]\left[ {1 + \sin (22.5)} \right]\left[ {1 - \sin (22.5)} \right]\\ = \left[ {1 - {{\cos }^2}(22.5)} \right]\left[ {1 - {{\sin }^2}(22.5)} \right]\\ = {\cos ^2}(22.5)si{n^2}(22.5)\\ = \cos (22.5)sin(22.5) \times \cos (22.5)sin(22.5)\\ = \frac{{2 \times \cos (22.5)sin(22.5)}}{2} \times \frac{{2 \times \cos (22.5)sin(22.5)}}{2}\\ = \frac{{sin(2 \times 22.5)}}{2} \times \frac{{sin(2 \times 22.5)}}{2}\\ = \frac{{sin(45)}}{2} \times \frac{{sin(45)}}{2}\\ = \frac{{\frac{{\sqrt 2 }}{2}}}{2} \times \frac{{\frac{{\sqrt 2 }}{2}}}{2}\\ = \frac{{\frac{2}{4}}}{4}\\ = \frac{1}{8} \end{array}

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