Prove that: 2cos π/13 cos 9π/13 + cos 3π/13 + cos 5π/13 = 0

Solution:

LHS = 2cos π/13 cos 9π/13 + [cos 3π/13 + cos 5π/13]

= 2cos π/13 cos 9π/13 + 2cos {(3π/13 + 5π/13) / 2} cos {(3π/13 - 5π/13) / 2} {Because cos A + cos B = 2cos [(A + B) / 2] cos [(A - B) / 2]}

= 2cos π/13 cos 9π/13 + 2cos 4π/13 cos (-π/13)

= 2cos π/13 cos 9π/13 + 2cos 4π/13 cos π/13   {Because cos (-x) = cos x}

= 2cos π/13(cos 9π/13 + cos 4π/13)

= 2cos π/13[2cos {(9π/13 + 4π/13) / 2} cos {(9π/13 - 4π/13) / 2}] {Because cos A + cos B = 2cos [(A + B) / 2] cos [(A - B) / 2]}

= 2cos π/13[2cos π/2 cos 5π/26]

= 2cos π/13 × 2 × 0 × cos 5π/26 [As cos π/2 = 0]

= 0

= RHS

NCERT Solutions Class 11 Maths Chapter 3 Exercise ME Question 1

Prove that: 2cos π/13 cos 9π/13 + cos 3π/13 + cos 5π/13 = 0

Summary:

We got, 2cos π/13 cos 9π/13 + cos 3π/13 + cos 5π/13 = 0. Hence Proved.