Question

The value of \( \frac{\sin^2 20^\circ + \cos^4 20^\circ}{\sin^4 20^\circ + \cos^2 20^\circ} \) is :

• A. 0
• B. 2
• C. 1
• D. \( \frac{1}{2} \)

Hint:

Try symmetry in trigonometric expressions—often numerator and denominator become identical.

Answer 1

The Correct Option is C

Step 1: Write numerator.
\[ \sin^2 20^\circ + \cos^4 20^\circ \]

Step 2: Write denominator.
\[ \sin^4 20^\circ + \cos^2 20^\circ \]

Step 3: Use identity.
\[ \sin^2 \theta + \cos^2 \theta = 1 \]

Step 4: Rewrite terms.
\[ \cos^4 20^\circ = (1 - \sin^2 20^\circ)^2 \]
\[ \sin^4 20^\circ = (1 - \cos^2 20^\circ)^2 \]

Step 5: Substitute and simplify both.
After simplification, numerator and denominator become equal.

Step 6: Hence ratio becomes.
\[ \frac{\text{same}}{\text{same}} = 1 \]

Step 7: Final Answer.
\[ \boxed{1} \]