The number of solutions of the equation cos2θcos(θ/2) + cos(5θ/2) = 2cos³(5θ/2) in [−π/2, π/2] is:
❓ Question Find the number of solutions of the equation cos 2 θ cos θ 2 + cos 5 θ 2 = 2 cos 3 5 θ 2 in the interval [ − π / 2 , π / 2 ]. 🖼️ Question Image ✍️ Short Solution Step 1 — Observe structure The RHS has 2 cos 3 5 θ 2 2\cos^3\frac{5\theta}{2} , which suggests the triple–angle identity: cos 3 x = 4 cos 3 x − 3 cos x . Step 2 — Rearrange equation Move RHS to LHS: cos 2 θ cos θ 2 + cos 5 θ 2 − 2 cos 3 5 θ 2 = 0. Write 2 cos 3 A = 1 2 ( 4 cos 3 A ) = 1 2 ( cos 3 A + 3 cos A ) 2\cos^3 A = \tfrac12 (4\cos^3A) = \tfrac12(\cos3A + 3\cos A) So: cos 2 θ cos θ 2 + cos 5 θ 2 − 1 2 ( cos 15 θ 2 + 3 cos 5 θ 2 ) = 0. Step 3 — Simplify LHS After simplification, the equation reduces to: cos 2 θ cos θ 2 + cos 5 θ 2 − 1 2 cos 15 θ 2 − 3 2 cos 5 θ 2 = 0. Collect like terms: cos 2 θ cos θ 2 − 1 2 cos 15 θ 2 − 1 2 cos 5 θ 2 = 0. Step 4 — Use product-to-sum cos 2 θ cos θ 2 = 1 2 [ cos ( 2 θ − θ 2 ) + cos ...