x=cos⁡55∘,y=cos⁡65∘,z=cos⁡175∘⇒xy+yz+zx=______

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$x = \cos 55^{\circ}, y = \cos 65^{\circ}, z = \cos 175^{\circ} \Rightarrow xy + yz + zx =$ _______

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answer is -0.75.

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Detailed Solution

$x + y + z = \cos 55 ° + \cos 65 ° + \cos 175 ° = 0 x^{2} + y^{2} + z^{2} = \cos^{2} 55 ° + \cos^{2} 65 ° + \cos^{2} 175 ° = \frac{3}{2} \Rightarrow {(x + y + z)}^{2} = 0 \Rightarrow x^{2} + y^{2} + z^{2} + 2 (x y + y z + z x) = 0 \Rightarrow \frac{3}{2} + 2 (x y + y z + z x) = 0 \Rightarrow x y + y z + z x = \frac{- 3}{4}$