The value of $$\cos^{2}3^{\circ} + \cos^{2}4^{\circ} + \cos^{2}5^{\circ} + \dots + \cos^{2}87^{\circ}$$ is:​

Correct option is D

Given:

$$\cos^2 3^\circ + \cos^2 4^\circ + \cos^2 5^\circ + \cdots + \cos^2 87^\circ$$​​

Formula Used:

​$$\cos^2 \theta + \cos^2 (90^\circ - \theta) = 1$$​​

​$$\cos^2 45^\circ = \frac{1}{2}$$​​

Solution:

The angles range from $$3^\circ$$ to $$87^\circ$$​, forming 85 - 3 + 1 = 85 terms.

Pair the terms symmetrically:

Each pair sums to 1.

Total pairs: $$\frac{84}{2}$$​ = 42 pairs (as the terms are symmetric).

the unpaired middle term: $$\cos^2 45^\circ = \frac{1}{2}$$​​

Total sum = $$42 \times 1 + \frac{1}{2} = \frac{84 + 1 }{2} = \frac{85}{2}$$​​