differentiate tan-1(√1+x2 -√1-x2/√1+x2 +√1-x2) with respect to cos-1x2 - Maths - Continuity and Differentiability

Question

differentiate tan-1(√1+x2
-√1-x2/√1+x2
+√1-x2) with respect to
cos​-1x2

Manbar Singh · Accepted Answer

Let u = tan-11+x2 - 1-x21+x2 + 1-x2Put x2 = cos 2θNow, u = tan-11+cos 2θ - 1 - cos 2θ1+cos 2θ + 1 - cos 2θ⇒u = tan-12 cos2θ - 2 sin2θ2 cos2θ + 2 sin2θ⇒u = tan-12 cos θ - 2 sin θ2 cos θ + 2 sin θ⇒u =  tan-1 cos θ - sin θ cos θ + sin θ⇒u =  tan-11 - tan θ1 + tan θ⇒u = tan-1tanπ/4 - tan θ1 + tanπ/4 × tan θ⇒u = tan-1tanπ4-θ⇒u =π4-θ⇒u = π4 - cos-1x22 ⇒dudx = 12×2x1-x4 = x1-x4Let v = cos-1x2⇒dvdx = -2x1-x4Now, dudv = dudx × dxdv = x1-x4 × -1-x42x = -12

differentiate tan-1(√1+x2 -√1-x2/√1+x2 +√1-x2) with respect to cos​-1x2

differentiate tan^-1(√1+x² -√1-x²/√1+x² +√1-x²) with respect to cos^-1x²