Answer :

To find the exact value of [tex]\(\cos 4A\)[/tex] given that [tex]\(\sin A \cos A = \frac{12}{25}\)[/tex], follow these steps:

1. Determine [tex]\(\sin 2A\)[/tex] and [tex]\(\cos 2A\)[/tex]:
- We use the double-angle identities:
[tex]\[
\sin 2A = 2 \sin A \cos A
\][/tex]
[tex]\[
\cos 2A = 1 - 2 \sin^2 A
\][/tex]
- Given [tex]\(\sin A \cos A = \frac{12}{25} \)[/tex]:
[tex]\[
\sin 2A = 2 \left( \frac{12}{25} \right) = \frac{24}{25}
\][/tex]

- To find [tex]\(\cos 2A\)[/tex], we can use another form of the double-angle identity for cosine:
[tex]\[
\cos 2A = \cos^2 A - \sin^2 A
\][/tex]
Using the Pythagorean identity [tex]\(\sin^2 A + \cos^2 A = 1\)[/tex], let [tex]\(\sin A = x\)[/tex] and [tex]\(\cos A = y\)[/tex], then:
[tex]\[
x^2 + y^2 = 1 \quad \text{and} \quad xy = \frac{12}{25}
\][/tex]
Therefore,
[tex]\[
\cos 2A = 1 - 2 \left( \frac{12}{25} \right)^2 = 1 - 2 \left( \frac{144}{625} \right) = 1 - \frac{288}{625} = \frac{337}{625}
\][/tex]

2. Determine [tex]\(\cos 4A\)[/tex]:
- Use the double-angle identity for [tex]\(4A\)[/tex]:
[tex]\[
\cos 4A = 2 \cos^2 2A - 1
\][/tex]
- Now, we have [tex]\(\cos 2A = \frac{337}{625}\)[/tex]:
[tex]\[
\cos^2 2A = \left( \frac{337}{625} \right)^2 = \frac{113569}{390625}
\][/tex]
- Substituting this into the identity for [tex]\(\cos 4A\)[/tex]:
[tex]\[
\cos 4A = 2 \left( \frac{113569}{390625} \right) - 1 = \frac{227138}{390625} - 1 = \frac{227138 - 390625}{390625} = \frac{-163487}{390625}
\][/tex]

So, the exact value of [tex]\(\cos 4A\)[/tex] is:
[tex]\[
\cos 4A = -0.41852672
\][/tex]

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