Question

If \(cos43^{\circ}+sin43^{\circ} = k^3\), then \(cos2^{\circ} = \ldots\)

• A. \(-\frac{k^3}{\sqrt{2}}\)
• B. \(\frac{k^3}{\sqrt{2}}\)
• C. \(-\frac{k^3}{\sqrt{3}}\)
• D. \(\frac{k^3}{\sqrt{3}}\)

Hint:

Write \(\cos43^{\circ}+\sin43^{\circ}=\sqrt2\cos(43^{\circ}-45^{\circ})\).

Answer 1

The Correct Option is B

Step 1: Use complementary angles
$\sin43^{\circ}=\cos47^{\circ}$, so the sum is $\cos43^{\circ}+\cos47^{\circ}$.

Step 2: Sum to product
$\cos43^{\circ}+\cos47^{\circ}=2\cos45^{\circ}\cos2^{\circ}=\sqrt2\cos2^{\circ}$.
Setting this equal to $k^3$ gives $\cos2^{\circ}=k^3/\sqrt2$, option (B).

Final Answer:
$\cos2^{\circ}=k^3/\sqrt2$, option (B). \[ \boxed{\dfrac{k^3}{\sqrt2}} \]