Question

If tan-1{root (1+x2) - root(1-x2)}/{ root (1+x2) +root (1-x2)} =theta , then prove that x2 =sin 2 theta .

Answer 1

Let u = tan−1[1−x2√x]put x = cos θ   u = tan−1[1−cos2θ√cosθ]⇒u = tan−1[sin θcos θ]⇒u = tan−1[tan θ]⇒u = θ⇒u = cos−1x⇒dudx = −11−x2√

Let v = cos−1[2x1−x2‾‾‾‾‾‾√]put x = sin θso, v = cos−1[2 sin θ 1 − sin2θ‾‾‾‾‾‾‾‾‾‾‾√]⇒v = cos−1[2 sin θ cos θ]⇒v = cos−1[sin 2θ]⇒v = cos−1[cos(π2−2θ)]⇒v = π2−2θ⇒v = π2−2sin−1x⇒v = −21−x2√Now,dudv = dudx ×dxdv = −11−x2√×−1−x2√2 = 12