Question : The value of the expression $\cos ^2 45^{\circ}+\cos ^2 135^{\circ}+\cos ^2 225^{\circ}+\cos ^2 315^{\circ}$ is:
Option 1: $2$
Option 2: $\frac{1}{2}$
Option 3: $\frac{3}{2}$
Option 4: $1$
Correct Answer: $2$
Solution :
Here, we have :
$\cos^245^\circ = (\frac{1}{\sqrt{2}})^2 = \frac{1}{2}$
$\cos^2135^\circ = \cos^2(90^\circ + 45^\circ ) = -\sin^245^\circ = (-\frac{1}{\sqrt{2}})^2 = \frac{1}{2}$
$\cos^2225^\circ = \cos^2(180^\circ + 45^\circ ) = -\cos^245^\circ = (-\frac{1}{\sqrt{2}})^2 = \frac{1}{2}$
$\cos^2315^\circ = \cos^2(360^\circ -45^\circ ) = \cos^245^\circ = (\frac{1}{\sqrt{2}})^2 = \frac{1}{2}$
$\cos^245^\circ + \cos^2135^\circ + \cos^2225^\circ + \cos^2315^\circ = ( \frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}) = 2$
Hence, the correct answer is 2.
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