Question

if cos theta + cos squared theta equal to 1 then prove that sin to the power 12 theta + 3 sin to the power 10 theta + 3 sin to the power 8 theta + sin to the power 6 theta + 2 Sin to the power 4 theta + 2 Sin to the power 4 theta minus 2 equal to 1

Answer 1

cos A + cos2 A = 1 . . . (1)

cos A = 1 - cos2 A

cos A = sin2 A . . . (2)

now, to prove,

sin 12 A + 3 sin10 A + 3 sin8 A + sin6 A + 2 sin4 A + 2 sin2 A - 2 = 1

LHS = sin 12 A + 3 sin10 A + 3 sin8 A + sin6 A + 2 sin4 A + 2 sin2 A - 2

= [(sin4 A)3 + 3 sin6 A (sin4 A + sin2 A) + (sin2 A)3] + 2 (sin4 A + sin2 A - 1)

= (sin4 A + sin2 A)3 + 2 [sin4 A + cos A - 1]

{using (a + b)3 = a3 + b3 + 3ab (a + b) and using (2) from above}

= [(sin2 A)2 + sin2 A]3 + 2 [(sin2 A)2 + cos A - 1]

= [(cos A)2 + sin2 A]3 + 2 [(cos A)2 + cos A - 1]

{using (2) from above}

= (cos2 A + sin2 A)3 + 2 (cos2 A + cos A - 1)

= (1)3 + 2 (1 - 1)

{using cos2 A + sin2 A = 1 and (1) from above}

= 1 + 2 (0)

= 1 + 0

= 1

= RHS

Answer 2

Answer:

hope this will help you......