NDA / CDS / AFCAT - MCQ Practice Questions

To find the sum of cosines at complementary angles, we use the sum-to-product formula to simplify the expression.

Step 1: Identify the Relationship

Notice that 75° = 90° - 15°, making these complementary angles where cos 75° = sin 15°.

Step 2: Apply Sum-to-Product Formula

We use the cosine sum-to-product identity: $\cos A+\cos B=2\cos \left(\frac{A+B}{2}\right)\cos \left(\frac{A-B}{2}\right)$

Step 3: Substitute Standard Values

Insert the known trigonometric values: $\cos 45°=\frac{\sqrt{2}}{2}$ and $\cos 30°=\frac{\sqrt{3}}{2}$

**The answer is (A) $\frac{\sqrt{6}}{2}$

x = 2 + √3, so 1/x = 1/(2+√3) = 2-√3 (after rationalization). x + 1/x = 4. Therefore x² + 1/x² = (x + 1/x)² - 2 = 16 - 2 = 14.

By Vieta's formulas, for a cubic x³ + bx² + cx + d = 0, sum of roots = -b. Here, sum = -(-6) = 6.