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Chapter 9: Problem 21

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

Short Answer

Expert verified
Question: Prove that \(\cos 4A = 8\cos^4 A - 8\cos^2 A + 1\). Solution: We used the double angle formula to rewrite \(\cos 4A\) as: $$\cos 4A = 2\cos^2(2A) - 1$$ Then, we applied the double angle formula again to rewrite \(\cos^2(2A)\) as: $$\cos^2(2A) = (2\cos^2 A - 1)^2$$ By substituting and simplifying the expression, we showed that: $$\cos 4A = 8\cos^4 A - 8\cos^2 A + 1$$

Step by step solution

01

Rewrite \(\cos 4A\) using the double angle formula

The double angle formula is given by \(\cos 2A = 2\cos^2 A - 1\). Let's apply this formula to \(4A\): $$ \cos 4A = \cos 2(2A) = 2\cos^2 (2A) - 1 $$
02

Rewrite \(\cos^2(2A)\) using the double angle formula again

We can again use the double angle formula \(\cos 2A = 2\cos^2 A - 1\), and apply it to \(2A\): $$ \cos^2 (2A) = (2\cos^2 A - 1)^2 $$
03

Substitute \(\cos^2(2A)\) in the expression for \(\cos 4A\)

Replace \(\cos^2 (2A)\) in the expression for \(\cos 4A\) with the expression we got in step 2: $$ \cos 4A = 2(2\cos^2 A - 1)^2 - 1 $$
04

Simplify the expression for \(\cos 4A\)

Now, we will simplify the expression: \begin{align*} \cos 4A &= 2(2\cos^2 A - 1)^2 - 1 \\ &= 2(4\cos^4 A - 4\cos^2 A + 1) - 1 \\ &= 8\cos^4 A - 8\cos^2 A + 2 - 1 \\ &= 8\cos^4 A - 8\cos^2 A + 1 \end{align*} Thus, we have shown that \(\cos 4A = 8\cos^4 A - 8\cos^2 A + 1\).

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Most popular questions from this chapter

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