How can you simplify ${sin}^{4} a + {cos}^{4} a$ $sin^4a + cos^4a$ ?

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How can you simplify ${sin}^{4} a + {cos}^{4} a$ $sin^4a + cos^4a$ ?

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Answer 1 · 2017-07-17T15:15:02

${sin}^{4} a + {cos}^{4} a = 1 - 2 {sin}^{2} a {cos}^{2} a = 1 - \frac{1}{2} {sin}^{2} (2 a)$ $sin^4a + cos^4a = 1 - 2sin^2acos^2a = 1 - 1/2sin^2(2a)$ .

Explanation:

Note that

${({sin}^{2} a + {cos}^{2} a)}^{2} = {sin}^{4} a + 2 {sin}^{2} a {cos}^{2} a + {cos}^{4} a$ $(sin^2a + cos^2a)^2 = sin^4a + 2sin^2acos^2a + cos^4a$

We see that ${sin}^{2} x + {cos}^{2} x = 1$ $sin^2x + cos^2x = 1$ , so that

$1 = {sin}^{4} a + 2 {sin}^{2} a {cos}^{2} a + {cos}^{4} a$ $1 = sin^4a + 2sin^2acos^2a + cos^4a$

Therefore,

${sin}^{4} a + {cos}^{4} a = 1 - 2 {sin}^{2} a {cos}^{2} a$ $sin^4a + cos^4a = 1 - 2sin^2acos^2a$

Hopefully this helps!

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How can you simplify ${sin}^{4} a + {cos}^{4} a$ $sin^4a + cos^4a$ ?

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