Question

Prove that : (cot^3 θ.sin^3θ)/(cos θ + sin θ)^2 + (tan^3 θ.cos^2 θ)/(cosθ + sin θ)^2

Answer 1

plz give full question

its incomplete

Answer 2

Step-by-step explanation:

LHS = $\frac{\frac{cos^3A}{sin^3A} . sin^3A}{(cosA + sinA)^2} + \frac{\frac{sin^3A}{cos^3A} . cos^3A}{(cosA + sinA)^2}$

= $\frac{cos^3A + sin^3A}{(cosA + sinA)^2}$

= $\frac{(cosA + sinA)(cos^2A + sin^2A - cosAsinA)}{(cosA + sinA)(cosA + sinA)}$

= $\frac{cos^2A + sin^2A - cosAsinA}{cosA + sinA}$

= $\frac{1-cosA sinA}{cosA+sinA}$ ------------ (1)

RHS = $\frac{cosecAsecA-1}{cosecA+secA}$

= $\frac{\frac{1}{sinA} . \frac{1}{cosA} -1 }{\frac{1}{sinA} + \frac{1}{cosA}}$

= $\frac{\frac{1}{sinAcosA}-1 }{\frac{cosA + sinA}{sinAcosA} }$

= $\frac{1-sinAcosA}{sinAcosA} . \frac{sinAcosA}{cosA+sinA}$

= $\frac{1-sinAcosA}{cosA+sinA}$-------------- (2)

From (1) & (2)

LHS = RHS

Hence, proved.