Question

PROVE THAT LHS = RHS ​

Answer 1

(tan^3 X/(1 + tan^2 X)) + (cot^3 X/(1 + cot^2 X))

= (tan^3 X/sec^2 X) + (cot^3 X/cosec^2 X)

= (sin^3 X/cos X) + (cos^3 X/sin X)

= (1/(sin X * cos X))*(sin^4 X + cos^4 X)

= (1/(sin X * cos X))*((sin^2 X)^2 + (cos^2 X)^2)

[Using (a^2 + b^2) = (a + b)^2 - 2ab]

= (1/(sin X * cos X))*((sin^2 X + cos^2 X)^2 - 2*sin^2 X*cos^2 X)

= (1/(sin X * cos X))*(1 - 2*sin^2 X*cos^2 X)

[since sin^2 X + cos^2 X = 1]

= 1/(sin X * cos X) - (2*sin^2 X*cos^2 X)/(sin X * cos X)

= cosec X * sec X - 2 * sin X * cos X

Hence Proved.