Question

if sin a = 1/2 (a + 1/a ) prove Sin 3 a = -1/2 ( a³+ 1/ a³)

Answer 1

if sin a = 1/2 (a + 1/a ) prove Sin 3 a = -1/2 ( a³+ 1/ a³)

proof : given, sina = 1/2 (a + 1/a)

we know, sin3θ= 3sinθ - 4sin³θ

so, LHS = sin3a = 3sina - 4sin³a

= 3[1/2(a + 1/a)] - 4[1/2(a + 1/a)]³

= 3/2(a + 1/a) - 4/2³(a + 1/a)³

use formula,

(a + b)³ = a³ + b³ + 3ab(a + b)

= 3/2(a + 1/a) - 1/2[a³ + 1/a³ + 3a × 1/a(a + 1/a) ]

= 3/2(a + 1/a) - 1/(a³ + 1/a³) - 3/2(a + 1/a)

= -1/2 (a³ + 1/a³) = RHS

hence, sin3a = -1/2(a³ + 1/a³)