If (cosec θ – sin θ) = a 3 and (sec θ – cos θ) = b 3 , prove that a 2 b 2 (a 2 + b 2 ) = 1.

Question

Philoid · Accepted Answer

Given: (cosec θ – sin θ) = a³

(sec θ – cos θ) = b³

Since, a³ = (cosec θ – sin θ)

= (1/sin θ) – sin θ

=

= cos² θ/sin θ

Therefore, a² = (a³)^2/3 = (cos² θ/sin θ)^2/3

Also, b³ = sec θ – cos θ

= (1/cos θ) – cos θ

= (1 – cos² θ)/cos θ

= sin² θ/cos θ

Therefore, b² = (b³)^2/3 = (sin² θ/cos θ)^2/3

Now, consider the left – hand side:

a² b² (a² + b²) =

=

= sin² θ + cos² θ

= 1 = R.H.S.

Hence, proved.