Question

Sin2acos4a(1+cos2a)/(sin3a+sina)(cos3a+cos5a)

Answer 1

Продолжение к фото:
=cos4a * 2cos(a^2) * cos4a = 2cos (4a)^2 * cos (a^2)

Answer 2

To simplify the given expression, let's break it down step by step:

1. Recall the double-angle formula for sine: sin(2a) = 2sin(a)cos(a).
2. Apply the double-angle formula to sin(2a), resulting in 2sin(a)cos(a)cos(4a)(1+cos(2a))/(sin(3a)+sin(a))(cos(3a)+cos(5a)).
3. Observe the double-angle formula for cosine: cos(2a) = cos^2(a) - sin^2(a) = 1 - 2sin^2(a).
4. Replace cos(2a) with 1 - 2sin^2(a), giving us 2sin(a)cos(a)(1 - 2sin^2(a))/((sin(3a)+sin(a))(cos(3a)+cos(5a))).
5. Apply the sum-to-product identities: sin(x) + sin(y) = 2sin((x+y)/2)cos((x-y)/2) and cos(x) + cos(y) = 2cos((x+y)/2)cos((x-y)/2).
6. Utilize the sum-to-product identities to rewrite the denominator as 2sin(2a)cos(4a).
7. Simplify the expression further, yielding sin(a)(1 - 2sin^2(a))/sin(2a)cos(4a).
8. Simplify sin(2a) by using the double-angle formula: sin(2a) = 2sin(a)cos(a).
9. Substitute sin(2a) in the expression, resulting in sin(a)(1 - 2sin^2(a))/(2sin(a)cos(a))cos(4a).
10. Cancel out sin(a) in the numerator and denominator, leaving (1 - 2sin^2(a))/(2cos(a))cos(4a).

The simplified expression is (1 - 2sin^2(a))/(2cos(a))cos(4a).