Question

\(\cos (4A)=\)

• A. \(1+8\cos ^{2}A+8\cos ^{4}A\)
• B. \(1+8\cos ^{2}A-8\cos ^{4}A\)
• C. \(1-8\cos ^{2}A+8\cos ^{4}A\)
• D. \(1-8\cos ^{2}A-8\cos ^{4}A\)

Hint:

Apply the double angle formula twice.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
Use \(\cos 2\theta=2\cos^{2}\theta-1\) twice.

Step 2: First application
\(\cos 4A=2\cos^{2}2A-1\).

Step 3: Second application
\(\cos 2A=2\cos^{2}A-1\), so \(\cos 4A=2(2\cos^{2}A-1)^{2}-1\).

Step 4: Expand
\(=2(4\cos^{4}A-4\cos^{2}A+1)-1=8\cos^{4}A-8\cos^{2}A+1\).

Final Answer:
cos 4A equals 1 - 8cos^2 A + 8cos^4 A. \[ \boxed{\text{(C) }1-8\cos ^{2}A+8\cos ^{4}A} \]