Question

Sin⁴a+cos⁴a+2sin²a×cos²a​

Answer 1

(a + b)^2 = a^2 +2ab + b^2
sin^4(A) + cos^4(A)+2sin^2(A)*cos^2(A) =
= (sin^2(A) + cos^2(A))^2 = 1^2 = 1

Answer 2

To simplify the expression sin^4(a) + cos^4(a) + 2sin^2(a)cos^2(a), you can use trigonometric identities. One of the most useful identities for this expression is the Pythagorean identity:

sin^2(a) + cos^2(a) = 1

From this identity, you can solve for sin^2(a) or cos^2(a) and substitute it into your expression:

sin^2(a) = 1 - cos^2(a)

Now, rewrite the original expression with this substitution:

sin^4(a) + cos^4(a) + 2sin^2(a)cos^2(a)

= (1 - cos^2(a))^2 + cos^4(a) + 2(1 - cos^2(a))cos^2(a)

Now, expand and simplify:

= 1 - 2cos^2(a) + cos^4(a) + cos^4(a) + 2cos^2(a) - 2cos^4(a)

Now, combine like terms:

= 1 + 2cos^2(a) - 2cos^4(a)

So, the simplified expression is 1 + 2cos^2(a) - 2cos^4(a).