Question

2cos²asin²a+cos⁴a+sin⁴a

Answer 1

$2cos^2\alpha \cdot \:sin^2\alpha +cos^4\alpha +sin^4\alpha =\\\\= (\sin ^4 (\alpha )+\sin ^2 (\alpha )\cos ^2 (\alpha ) )+ (\sin ^2 (\alpha )\cos ^2 (\alpha )+\cos ^4 (\alpha ) )=\\\\=\sin ^2 (\alpha ) (\sin ^2 (\alpha )+\cos ^2 (\alpha ) )+\cos ^2 (\alpha ) (\sin ^2 (\alpha )+\cos ^2 (\alpha ) )=\\\\= (\sin ^2 (\alpha )+\cos ^2 (\alpha ) ) (\sin ^2 (\alpha )+\cos ^2 (\alpha ) )=\\\\= (\sin ^2 (\alpha )+\cos ^2 (\alpha ) )^2\\\\=1^2=1$

Answer 2

To simplify the expression 2cos²(a)sin²(a) + cos⁴(a) + sin⁴(a), we can use some trigonometric identities. One of the most useful identities is the Pythagorean identity, which states that sin²(a) + cos²(a) = 1.

Let's start by rewriting the expression using the Pythagorean identity:

cos⁴(a) + sin⁴(a) = (cos²(a) + sin²(a))(cos²(a) - sin²(a))

Now, replace cos²(a) with (1 - sin²(a)):

(cos²(a) + sin²(a))(cos²(a) - sin²(a)) = (1 - sin²(a))(cos²(a) - sin²(a))

Now, we can use the difference of squares identity: a² - b² = (a + b)(a - b)

So, apply the identity:

(1 - sin²(a))(cos²(a) - sin²(a)) = (1 - sin²(a))(cos(a) + sin(a))(cos(a) - sin(a))

Now, let's focus on the other term in the original expression:

2cos²(a)sin²(a) = 2 * (cos(a)sin(a))^2

We can use the double angle identity for sine: sin(2a) = 2sin(a)cos(a)

Therefore, cos(a)sin(a) = 0.5sin(2a)

Substitute this back into the expression:

2 * (cos(a)sin(a))^2 = 2 * (0.5sin(2a))^2 = 0.5^2 * 2 * sin^2(2a) = 0.5 * sin^2(2a)

Now, we have:

2cos²(a)sin²(a) + cos⁴(a) + sin⁴(a) = 0.5 * sin²(2a) + (1 - sin²(a))(cos(a) + sin(a))(cos(a) - sin(a))

Now, you can leave it in this form, or if you want to simplify it further, it will depend on the context and what you're trying to achieve with the expression.