Question

Evaluate : \(\sin^2 30^\circ - \cos^2 45^\circ + \cot^2 60^\circ\)

Hint:

When an equation involves \(\sin(\dots) = 1\), immediately replace 1 with \(\sin 90^\circ\) to "remove" the sine function and solve for the angles.

Answer 1

Step 1: Understanding the Concept:
Evaluation involves substituting the standard trigonometric values for specific angles into the expression.
Step 2: Key Values:
\(\sin 30^\circ = \frac{1}{2}\), \(\cos 45^\circ = \frac{1}{\sqrt{2}}\), \(\cot 60^\circ = \frac{1}{\sqrt{3}}\).
Step 3: Detailed Explanation:
1. Substitute the values:
\[ \left(\frac{1}{2}\right)^2 - \left(\frac{1}{\sqrt{2}}\right)^2 + \left(\frac{1}{\sqrt{3}}\right)^2 \] 2. Square the terms:
\[ \frac{1}{4} - \frac{1}{2} + \frac{1}{3} \] 3. Find a common denominator (12):
\[ \frac{3 - 6 + 4}{12} = \frac{1}{12} \]
Step 4: Final Answer:
The value is \(\frac{1}{12}\).