The value of $c o s^{2} 10^{o} - c o s 10^{o} c o s 50^{o} + c o s^{2} 10^{o}$ is equal to

\frac{4}{3}

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\frac{1}{3}

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\frac{3}{4}

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Solution

The correct option is C $\frac{3}{4}$
Given, $c o s^{2} 10^{o} - c o s 10^{o} c o s 50^{o} + c o s^{2} 10^{o}$

$= \frac{1}{2} (2 c o s^{2} 10^{o} - 2 c o s 10^{o} c o s 50^{o} + 2 c o s^{2} 50^{o})$
$= \frac{1}{2} (1 + c o s 20^{o} - (c o s 60^{o} + c o s 40^{o}) + 1 + c o s 100^{o})$
$= \frac{1}{2} (2 + c o s 20^{o} - \frac{1}{2} - c o s 40^{o} + c o s (180^{o} - 80^{o}))$
$= \frac{1}{2} (\frac{3}{2} + c o s 20^{o} - (c o s 80^{o} + c o s 40^{o}))$
$= \frac{1}{2} (\frac{3}{2} + c o s 20^{o} - (2 c o s 60^{o} c o s 20^{o}))$
$= \frac{1}{2} (\frac{3}{2} + c o s 20^{o} - (2 (\frac{1}{2}) c o s 20^{o})$
$= \frac{1}{2} (\frac{3}{2} + c o s 20^{o} - c o s 20^{o})$
$= \frac{1}{2} (\frac{3}{2})$
$= \frac{3}{4}$

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