Question

Assertion (A): \( cos^{2}5^{\circ}+cos^{2}10^{\circ}+cos^{2}15^{\circ}+...+cos^{2}85^{\circ}=\frac{17}{2} \)
Reason (R): If \( A+B=90^{\circ} \), then \( cos^{2}A+cos^{2}B=1 \)
Then, which one of the following is True?

• A. A is true, R is true and R is the correct explanation of A
• B. A is true, R is true and R is not correct explanation of A
• C. A is true, R is false
• D. A is false, R is true

Hint:

For any finite symmetric trigonometric series of squares where \( A+B=90^{\circ} \), the total sum can be quickly written as \( \frac{\text{Total number of terms}}{2} \). Here, \( \frac{17}{2} \) can be written directly without any grouping steps!

Answer 1

The Correct Option is A

Step 1: Check the reason first.
If $A+B=90^\circ$, then $B = 90^\circ - A$, so $\cos^2 B = \sin^2 A$. Then $\cos^2 A + \cos^2 B = \cos^2 A + \sin^2 A = 1$. So the reason is true.
Step 2: Count the terms.
The angles $5^\circ, 10^\circ, \ldots, 85^\circ$ go in steps of 5. The count is $\dfrac{85-5}{5}+1 = 17$ terms.
Step 3: Pair complementary angles.
Pair the first with the last: $\cos^2 5^\circ + \cos^2 85^\circ = 1$, and so on, since each pair adds to $90^\circ$.
Step 4: Find the leftover term.
With 17 terms, there are 8 full pairs and one middle term, $\cos^2 45^\circ = \left(\tfrac{1}{\sqrt2}\right)^2 = \tfrac{1}{2}$.
Step 5: Add it all.
Total $= 8\times 1 + \tfrac{1}{2} = \tfrac{17}{2}$. So the assertion is true.
Step 6: Decide the link.
The pairing used the reason directly, so the reason correctly explains the assertion. \[ \boxed{\text{A true, R true, R explains A}} \]