trigonometry

(tan³θ / 1+tan²θ)  +( cot ³θ / 1+cot²θ ) =

(tan³θ / sec²θ)  +( cot ³θ / cosec²θ ) =

(sin³θ / cos θ)  +( cos ³θ / sin θ ) =

(sin⁴θ + cos ⁴θ)/(sin θcos θ) ..........................(1)

Now, (sin²θ + cos ²θ)² = sin⁴θ + cos ⁴θ + 2sin² θcos² θ

Therefore, sin⁴θ + cos ⁴θ = (sin²θ + cos ²θ)²- 2sin² θcos² θ = 1 - 2sin² θcos² θ

Substituting this in (1),

(1 - 2sin² θcos² θ)/(sin θcos θ) =

sec θ cosec θ - 2sin θ cos θ

Regards,

Team,

TopperLearning.