Introduction / Context:
This problem checks your ability to recall and work with basic trigonometric ratios for standard angles and to perform simple operations with surds. The expression involves both Cot and Cosec functions at commonly used angles 45 degrees and 60 degrees.
Given Data / Assumptions:
- Angles are in degrees.
- Expression: Cot 45° − (1 / √3) * Cosec 60°.
- Standard values for trigonometric functions at 45° and 60° are known.
Concept / Approach:
We recall that Cot 45° = 1 and Sin 60° = √3 / 2. Thus Cosec 60° is the reciprocal of Sin 60°. After substituting the standard values, we simplify the combination by careful handling of surds and fractions to get a final exact result.
Step-by-Step Solution:
Recall Cot 45° = 1.
Recall Sin 60° = √3 / 2, so Cosec 60° = 1 / Sin 60° = 2 / √3.
Substitute into the expression: Cot 45° − (1 / √3) * Cosec 60° = 1 − (1 / √3) * (2 / √3).
Compute the second term: (1 / √3) * (2 / √3) = 2 / 3.
So the expression becomes 1 − 2 / 3.
Compute the difference: 1 − 2 / 3 = 1 / 3.
Verification / Alternative check:
Using approximate decimals, Cot 45° is 1. Cosec 60° is 2 / √3 which is about 1.155. Then (1 / √3) * Cosec 60° is approximately 0.577 * 1.155 which is about 0.667. So the whole expression is about 1 − 0.667 = 0.333, which matches 1 / 3.
Why Other Options Are Wrong:
Option a (1 / √3) is about 0.577, which is too large. Option b (1 / 2) is about 0.5, still larger than the correct value. Option c (1 / √2) is about 0.707. Option e (0) would require the two terms to cancel exactly, which they do not. Only 1 / 3 matches the exact computation.
Common Pitfalls:
Learners may mistakenly use Cosec 45° instead of Cosec 60°, or confuse Cot 45° with Tan 45°. Errors also occur when multiplying surds and simplifying denominators. Carefully substituting the correct values and simplifying step by step avoids these mistakes.
Final Answer:
The exact value of Cot 45° − (1 / √3) Cosec 60° is
1 / 3.
Discussion & Comments