Question

prove that ................​

Answer 1

Taking LHS

(Tan^3∅)/(1+tan^2∅) +(cot^3∅)/(1+cot^2∅)

=>(Tan^3∅)/(1+Tan^2∅) +(cot^3∅)/(1+1/tan^2∅)

=>(Tan^3∅)/(1+tan^2∅) +(tan^2∅cot^3∅)/(tan^2∅+1)

now,1+Tan^2∅ is denominator in both term

=>(Tan^3∅+Tan^2∅/Tan^3∅)/(Tan^2∅+1)

=>(Tan^3∅+1/Tan∅)/(sec^2∅)

converting it to sin and cos term

=>(sin^3∅/cos^3∅ +cos∅/sin∅)/(sec^2∅)

=>(Sin^4∅+ cos^4∅)/(cos^3∅sin∅×sec^2∅)

=>(Sin^4∅+cos^4∅)/(cos∅sin∅)

=>{(sin^2∅)^2+ (cos^2∅)^2+2sin^2∅cos^2∅-2sin^2∅cos^2∅)}/(cos∅sin∅)

=>{(sin^∅+cos^2∅)^2-2sin^2∅cos^2∅}/(cos∅sin∅)

=>(1-2sin^2∅cos^2∅)/(cos∅sin∅)

=>sec∅cosec∅-2sin∅cos∅=RHS

Hence proved

{hope it helps}