If x = a cosec θ sin ϕ, y = b cosec θ cos ϕ and z = c cot θ then prove that (x2a2+y2b2)=(1+z2c2).
Answer:
- We are given that xa=cosec θ sin ϕ…(i)yb=cosec θ cos ϕ…(ii)zc=cot θ…(iii)
- On squaring and adding (i) and (ii), we get (x2a2+y2b2)=cosec2 θ(sin2 ϕ+cos2 ϕ)=cosec2 θ[∵sin2 ϕ+cos2 ϕ=1]=(1+cot2 θ)[∵cosec2=1+cot2 θ]=(1+z2c2)[By equation (iii)]
- Hence,(x2a2+y2b2)=(1+z2c2).