Question

\(\frac{\cos 59^\circ}{\sin 31^\circ} \times \frac{\tan 80^\circ}{\cot 10^\circ} = \)

• A. \(\frac{1}{\sqrt{2}}\)
• B. 1
• C. \(\frac{\sqrt{3}}{2}\)
• D. \(\frac{1}{2}\)

Hint:

When you see a fraction with a trigonometric function and its co-function (sin/cos, tan/cot, sec/csc), check if the angles add up to 90°. If they do, the value of the fraction is 1.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This problem can be simplified using complementary angle identities, which relate a trigonometric function of an angle to the co-function of its complement.

Step 2: Key Formula or Approach:
The key identities are:
\[ \cos \theta = \sin(90^\circ - \theta) \] \[ \tan \theta = \cot(90^\circ - \theta) \]

Step 3: Detailed Explanation:
Let's simplify each fraction separately.
First fraction: \(\frac{\cos 59^\circ}{\sin 31^\circ}\)
Using the identity, we can rewrite the numerator: \(\cos 59^\circ = \sin(90^\circ - 59^\circ) = \sin 31^\circ\).
So, the fraction becomes \(\frac{\sin 31^\circ}{\sin 31^\circ} = 1\).
Second fraction: \(\frac{\tan 80^\circ}{\cot 10^\circ}\)
Using the identity, we can rewrite the numerator: \(\tan 80^\circ = \cot(90^\circ - 80^\circ) = \cot 10^\circ\).
So, the fraction becomes \(\frac{\cot 10^\circ}{\cot 10^\circ} = 1\).
The entire expression is the product of the two simplified fractions:
\[ 1 \times 1 = 1 \]

Step 4: Final Answer:
The value of the expression is 1.