Question

Evaluate: \[ \cos^2 45^\circ+\cos^2 135^\circ+\cos^2 225^\circ+\cos^2 315^\circ= \]

• A. \(1\)
• B. \(2\)
• C. \(0\)
• D. \(-1\)

Hint:

Even if cosine values become negative in different quadrants, \[ \cos^2\theta \] always remains non-negative.

Answer 1

The Correct Option is B

Step 1: Find \(\cos 45^\circ\).
We know that \[ \cos 45^\circ=\frac{1}{\sqrt{2}} \] Therefore, \[ \cos^2 45^\circ=\frac{1}{2} \]

Step 2: Find \(\cos 135^\circ\).
Since \[ 135^\circ=180^\circ-45^\circ, \] we get \[ \cos 135^\circ=-\frac{1}{\sqrt{2}} \] Thus, \[ \cos^2 135^\circ=\frac{1}{2} \]

Step 3: Find \(\cos 225^\circ\).
Since \[ 225^\circ=180^\circ+45^\circ, \] we get \[ \cos 225^\circ=-\frac{1}{\sqrt{2}} \] Therefore, \[ \cos^2 225^\circ=\frac{1}{2} \]

Step 4: Find \(\cos 315^\circ\).
Since \[ 315^\circ=360^\circ-45^\circ, \] we get \[ \cos 315^\circ=\frac{1}{\sqrt{2}} \] Hence, \[ \cos^2 315^\circ=\frac{1}{2} \]

Step 5: Add all the values.
Now, \[ \cos^2 45^\circ+\cos^2 135^\circ+\cos^2 225^\circ+\cos^2 315^\circ \] \[ = \frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2} \] \[ =2 \]

Step 6: Match with the options.
The obtained value is \[ 2 \]

Step 7: Final conclusion.
Therefore, \[ \boxed{2} \]