Question

If x=p cos 2 Ɵ + q sin 2 Ɵ, then show that (x-p) (q-x) = (p-q) 2 sin 2 Ɵcos 2 Ɵ

Answer 1

x=pcos²θ+qsin²θ
∴, (x-p)(q-x)
=(pcos²θ+qsin²θ-p)(q-pcos²θ-qsin²θ)
={p(cos²θ-1)+qsin²θ}{q(1-sin²θ)-pcos²θ}
={p(-sin²θ)+qsin²θ}(qcos²θ-pcos²θ) [∵, sin²θ+cos²θ=1]
={sin²θ(q-p)}{cos²θ(q-p)}
={-(p-q)}{-(p-q)}sin²θcos²θ
=(p-q)²sin²θcos²θ (Proved)