Question

prove that: 4sin^3a.cos3a + 4cos^3a.sin3a = 3sin4a

Answer 1

Answer:

Step-by-step explanation:

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Answer 2

$4sin^3a cos3a + 4cos^3asin3a = 3sin4a$

Solution:

Given that, we have to prove:

$4sin^3a cos3a + 4cos^3asin3a = 3sin4a$

Take the LHS

$4sin^3a cos3a + 4cos^3asin3a$ ---- ( 1)

We know that,

$cos\ 3a = 4cos^3a-3cos\ a$

$sin\ 3a = 3sin\ a -4sin^3a$

Apply these in (1)

$4sin^3a ( 4cos^3a-3cos\ a) + 4cos^3a(3sin\ a -4sin^3a)$

Expand

$16sin^3acos^3a - 12sin^3acosa+12cos^3asina - 16cos^3asin^3a\\\\Cancel\\\\- 12sin^3acosa+12cos^3asina\\\\Take\ 12sinacosa\ as\ common\\\\12sin\ a\ cos\ a(cos^2a - sin^2a)\\\\We\ know\ that\ cos^2a - sin^2a = cos2a\\\\Therefore\\\\12sin\ a\ cos\ a\ cos2a\\\\6(2sin\ a\ cos\ a)cos\ 2a\\\\6sin\ 2a\ cos\ 2a\\\\3(2sin2a\ cos\ 2a)\\\\3sin4a$

Thus,

LHS = RHS

Thus proved

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