Show that cos 4 alpha =8cos^4alpha - 8cos^2alpha + 1

Answer 1

To prove that cos(4α) = 8cos^4(α) - 8cos^2(α) + 1, we can use the double angle formula for cosine and a trigonometric identity.

The double angle formula for cosine states that:
cos(2θ) = 2cos^2(θ) - 1

Let's substitute α for θ in this formula:
cos(2α) = 2cos^2(α) - 1

Now we can use the trigonometric identity for cosine to express cos(4α) in terms of cos(2α):
cos(4α) = cos(2(2α))

Using the double angle formula again:
cos(4α) = 2cos^2(2α) - 1

Now we need to express cos^2(2α) in terms of cos^2(α). We can use another trigonometric identity for cosine:
cos(2θ) = 1 - 2sin^2(θ)

Using this identity, we can express cos^2(2α) in terms of sin^2(α):
cos^2(2α) = 1 - 2sin^2(α)

Now we substitute this expression back into the previous equation:
cos(4α) = 2(1 - 2sin^2(α)) - 1
cos(4α) = 2 - 4sin^2(α) - 1
cos(4α) = 1 - 4sin^2(α)
cos(4α) = 4(1 - sin^2(α)) - 3
cos(4α) = 4cos^2(α) - 3

Finally, we use the trigonometric identity cos^2(α) = 1 - sin^2(α):
cos(4α) = 4(1 - sin^2(α)) - 3
cos(4α) = 4cos^2(α) - 3

Comparing this equation with the one we need to prove, we see that they are equal:
cos(4α) = 4cos^2(α) - 3 = 8cos^4(α) - 8cos^2(α) + 1

Therefore, we have shown that cos(4α) = 8cos^4(α) - 8cos^2(α) + 1.