Question

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

Short answer

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

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 \(4A\): $$ \cos 4A = \cos 2(2A) = 2\cos^2 (2A) - 1 $$

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 \(2A\): $$ \cos^2 (2A) = (2\cos^2 A - 1)^2 $$

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: $$ \cos 4A = 2(2\cos^2 A - 1)^2 - 1 $$

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

Key concepts

Understanding the Double Angle Formula

The double angle formula is a critical tool in trigonometry. It helps in simplifying expressions where the angle is doubled. When you hear about the double angle formula, think of it as a way to express trigonometric functions of double angles, like \(2A\), in terms of single angles. This makes calculations and simplifications easier.

The most commonly used form of the double angle formula for cosine is:

• \(\cos 2A = 2\cos^2 A - 1\)

This tells us that the cosine of double an angle is equal to twice the square of the cosine of the single angle, subtracted by 1.

In our exercise, this formula is crucial because it helps us express \(\cos 4A\) in terms of \(A\). By repeatedly applying the double angle formula, we can break down the problem into simpler steps.

Exploring the Cosine Function

The cosine function is fundamental in trigonometry, representing the relationship between the angles and sides of a right triangle. When dealing with angles such as \(4A\) and \(2A\), understanding \(\cos A\) is essential because it becomes the foundation for more complex expressions.

When simplifying expressions involving \(\cos 4A\), knowing how cosine behaves when you double, and even quadruple, the angle helps build intuition. The cosine function's intrinsic properties, like its symmetry and periodicity, make it predictable, allowing derivation of complex trigonometric identities from base angles.

Moreover, cosine values for different angles, especially angles that are multiples of a base angle, are interconnected through identities like the double angle formula. Such identities unify different expressions and showcase how versatile and interconnected trigonometric functions are.

Mastering Algebraic Manipulation

Algebraic manipulation is like playing with puzzle pieces, where you rearrange and combine parts to simplify or solve equations. It often involves expanding expressions, factoring them, or substituting equivalent expressions to make complex problems more manageable.

In the problem \(\cos 4A = 8\cos^4 A - 8\cos^2 A + 1\), algebraic manipulation takes center stage after applying trigonometric identities. Once the double angle formula translates \(\cos 4A\) into a usable form, the real work begins.

• First, expand \((2\cos^2 A - 1)^2\), giving us \(4\cos^4 A - 4\cos^2 A + 1\).
• Then, substitute this expression into the equation for \(\cos 4A\) and simplify.
• Combining like terms and following arithmetic operations finally yields the expression \(8\cos^4 A - 8\cos^2 A + 1\).

Each algebraic step must be precise, as even small missteps can lead to errors. Through careful manipulation, you transform complicated expressions into verifications of identities.