Introduction / Context:
This trigonometry question checks your recall of exact values of trigonometric functions for special angles such as 30 degrees, 45 degrees, and 60 degrees. You are asked to compute cot 45 degrees plus cosec 60 degrees and express the answer as a simplified surd fraction. Such problems are common in aptitude tests because they reward memorisation of special angle values and basic algebraic manipulation.
Given Data / Assumptions:
- Angle 45 degrees is a standard special angle.
- Angle 60 degrees is also a standard special angle.
- cot 45 degrees = 1.
- cosec θ = 1 / sin θ for any angle θ where sine is defined.
- sin 60 degrees = √3/2 for the principal non negative value.
Concept / Approach:
The strategy is to use the known exact values directly. First, evaluate cot 45 degrees. Then, find sin 60 degrees and invert it to get cosec 60 degrees. Finally, add the two values and simplify the result so that it matches one of the given options. No calculator is needed as long as the special angle values are memorised correctly.
Step-by-Step Solution:
Recall that tan 45 degrees = 1, so cot 45 degrees, which is 1 / tan 45 degrees, also equals 1.
Recall sin 60 degrees = √3/2.
Therefore cosec 60 degrees = 1 / sin 60 degrees = 1 / (√3/2) = 2/√3.
Now compute the sum: cot 45 degrees + cosec 60 degrees = 1 + 2/√3.
Express 1 with denominator √3: 1 = √3/√3.
Add the fractions: √3/√3 + 2/√3 = (√3 + 2)/√3.
Verification / Alternative check:
You can perform a quick numerical check to confirm. Approximate √3 as about 1.732. Then cosec 60 degrees = 2/√3 is approximately 1.155, and cot 45 degrees is exactly 1. Their sum is roughly 2.155. Evaluating (√3 + 2)/√3 numerically yields (1.732 + 2)/1.732 ≈ 3.732/1.732 ≈ 2.155, which matches the earlier approximate sum and confirms the expression is correct.
Why Other Options Are Wrong:
Option b ( (√6 + 1)/√3 ) and option d ( (1 + √3)/2 ) represent different combinations of surds and fractions that do not match the exact sum of 1 and 2/√3. Option c ( 5/√3 ) would require the sum to be 5/√3 ≈ 2.886, which is larger than the actual value. Option e ( (2 + √3)/2 ) approximates to a value near 1.866 and therefore does not equal the computed sum.
Common Pitfalls:
A common mistake is to confuse sine and cosine values at 60 degrees, for example using sin 60 degrees = 1/2 and cos 60 degrees = √3/2 instead of the correct values. Another error is to forget that cosec is the reciprocal of sine, not cosine, leading to an incorrect 2/1 or 1/2. When simplifying the final expression, some learners also forget to put 1 over a common denominator before adding fractions, which leads to mismatched forms compared to the options.
Final Answer:
The exact simplified value of cot 45 degrees + cosec 60 degrees is
(√3 + 2)/√3.
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