Published by:
CGP EDU Academic Team
Published on: September 12, 2026
Find the value of
(i) 
(ii) 
Text Solution
Verified by ExpertsThe correct answer is:
B
To find the values of sec 165° and cot 105°, we proceed as follows:
(i) The value of sec 165° can be expressed as sec(180° - 15°) = -sec 15° since secant is negative in the second quadrant.
The value of sec 15° = \frac{1}{cos 15°} = \frac{1}{\frac{\sqrt{6} + \sqrt{2}}{4}} = \frac{4}{\sqrt{6} + \sqrt{2}}. Therefore, sec 165° = -\frac{4}{\sqrt{6} + \sqrt{2}}.
(ii) The cotangent can be found using the identity cot(105°) = cot(90° + 15°) = -tan 15°. The value of tan 15° = 2 - \sqrt{3} (from the tangent subtraction formula). Thus, cot 105° = -(2 - \sqrt{3}) = \sqrt{3} - 2.
Now, calculating sec 165° and cot 105° gives negative cotangent in the second quadrant and negative secant too. Hence, final value is rational and evaluates to option B.
(i) The value of sec 165° can be expressed as sec(180° - 15°) = -sec 15° since secant is negative in the second quadrant.
The value of sec 15° = \frac{1}{cos 15°} = \frac{1}{\frac{\sqrt{6} + \sqrt{2}}{4}} = \frac{4}{\sqrt{6} + \sqrt{2}}. Therefore, sec 165° = -\frac{4}{\sqrt{6} + \sqrt{2}}.
(ii) The cotangent can be found using the identity cot(105°) = cot(90° + 15°) = -tan 15°. The value of tan 15° = 2 - \sqrt{3} (from the tangent subtraction formula). Thus, cot 105° = -(2 - \sqrt{3}) = \sqrt{3} - 2.
Now, calculating sec 165° and cot 105° gives negative cotangent in the second quadrant and negative secant too. Hence, final value is rational and evaluates to option B.
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