The value of \(\frac{cos^6θ + sin^6θ + 3sin^2θ cos^2θ}{cosec θ sec θ ( 1 - sin θ - cos θ) ( 1+ sin θ + cos θ)}\) is ?

-\(\frac{1}{2}\)

Here,

⇒ cos⁶ θ + sin⁶ θ + 3sin² cos² θ

= cos⁶ θ + sin⁶ θ + 3sin² cos² θ (sin² θ + cos² θ)

= (cos² θ + sin² θ)³ = 1

Now,

= \(\frac{1}{cosec θ sec θ ( 1 - (sin θ + cos θ)(1 + (sin θ + cos θ)}\)

= \(\frac{1}{cosec θ sec θ (1^2 - (sinθ + cosθ)^2)}\)

= \(\frac{1}{cosec θ sec θ ( - 2sin θ cos θ) }\)

= - \(\frac{1}{2}\)