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?
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?
Reason (R): If \( A+B=90^{\circ} \), then \( cos^{2}A+cos^{2}B=1 \)
Then, which one of the following is True?
Show 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!
Updated On: Jun 8, 2026
- A is true, R is true and R is the correct explanation of A
- A is true, R is true and R is not correct explanation of A
- A is true, R is false
- A is false, R is true
Show Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
Concept:
Let us verify the Reason (R) first. If \( A + B = 90^{\circ} \), then \( B = 90^{\circ} - A \).
\[
cos^{2}B = cos^{2}(90^{\circ} - A) = sin^{2}A
\]
Thus, \( cos^{2}A + cos^{2}B = cos^{2}A + sin^{2}A = 1 \). This proves that the Reason (R) is true.
Step 1: Counting and grouping the terms in Assertion (A).
The series is \( cos^{2}5^{\circ} + cos^{2}10^{\circ} + cos^{2}15^{\circ} + \dots + cos^{2}85^{\circ} \). The angles form an Arithmetic Progression: \( 5, 10, 15, \dots, 85 \). The total number of terms \( n \) is given by: \[ n = \frac{85 - 5}{5} + 1 = \frac{80}{5} + 1 = 17 \text{ terms} \]
Step 2: Pairing complementary angles together.
Using the property from Reason (R), we can pair the first and last terms, second and second-to-last terms, etc., since their angles sum up to \( 90^{\circ} \):
• \( cos^{2}5^{\circ} + cos^{2}85^{\circ} = 1 \)
• \( cos^{2}10^{\circ} + cos^{2}80^{\circ} = 1 \)
• \( \dots \)
Out of 17 terms, there are exactly 8 complete pairs, leaving out one single unpaired middle term: \[ \text{Middle term} = cos^{2}45^{\circ} = \left(\frac{1}{\sqrt{2}}\right)^{2} = \frac{1}{2} \]
Step 3: Summing the full expression.
\[ \text{Total Sum} = (8 \times 1) + \frac{1}{2} = 8 + \frac{1}{2} = \frac{17}{2} \] Thus, Assertion (A) is true, and it is directly explained by the pairing rule stated in Reason (R).
Step 1: Counting and grouping the terms in Assertion (A).
The series is \( cos^{2}5^{\circ} + cos^{2}10^{\circ} + cos^{2}15^{\circ} + \dots + cos^{2}85^{\circ} \). The angles form an Arithmetic Progression: \( 5, 10, 15, \dots, 85 \). The total number of terms \( n \) is given by: \[ n = \frac{85 - 5}{5} + 1 = \frac{80}{5} + 1 = 17 \text{ terms} \]
Step 2: Pairing complementary angles together.
Using the property from Reason (R), we can pair the first and last terms, second and second-to-last terms, etc., since their angles sum up to \( 90^{\circ} \):
• \( cos^{2}5^{\circ} + cos^{2}85^{\circ} = 1 \)
• \( cos^{2}10^{\circ} + cos^{2}80^{\circ} = 1 \)
• \( \dots \)
Out of 17 terms, there are exactly 8 complete pairs, leaving out one single unpaired middle term: \[ \text{Middle term} = cos^{2}45^{\circ} = \left(\frac{1}{\sqrt{2}}\right)^{2} = \frac{1}{2} \]
Step 3: Summing the full expression.
\[ \text{Total Sum} = (8 \times 1) + \frac{1}{2} = 8 + \frac{1}{2} = \frac{17}{2} \] Thus, Assertion (A) is true, and it is directly explained by the pairing rule stated in Reason (R).
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