Question

The value of cos 52° + cos 68° + cos 172° is

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Answer 1

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\[\cos52^\circ + \cos68^\circ + \cos172^\circ\]
\[ = 2\cos\left( \frac{52^\circ + 68^\circ}{2} \right)\cos\left( \frac{52^\circ - 68^\circ}{2} \right) + \cos172^\circ \left[ \because \cos A + \cos B = 2\cos\left( \frac{A + B}{2} \right)\cos\left( \frac{A - B}{2} \right) \right]\]
\[ = 2\cos60^\circ\cos\left( - 8^\circ \right) + \cos172^\circ\]
\[ = 2 \times \frac{1}{2}\cos8^\circ + \cos172^\circ\]
\[ = \cos8^\circ + \cos172^\circ\]
\[ = 2\cos\left( \frac{8^\circ + 172^\circ}{2} \right)\cos\left( \frac{8^\circ - 172^\circ}{2} \right)\]
\[ = 2\cos90^\circ\cos82^\circ\]
\[ = 0\]