$\left(\sqrt{\sec ^2 \theta+{cosec}^2 \theta}\right)\left(\frac{\sin \theta(1+\cos \theta)}{1+\cos \theta-\sin ^2 \theta}\right), 0^{\circ}<\theta<90^{\circ}$ is equal to:

$\sec ^2 \theta$

( \(\sqrt {sec²θ + cosec²θ }\) ) . ( \(\frac{ sinθ ( 1 + cosθ)}{1 + cosθ - sin²θ}\) )

{ Using ,  secθ = \(\frac{1}{cosθ}\) , cosecθ = \(\frac{1}{sinθ}\) & sin²θ + cos²θ = 1 }

= ( \(\sqrt {1/cos²θ + 1/sin²θ }\) ) . ( \(\frac{ sinθ ( 1 + cosθ)}{1 + cosθ - 1 + cos²θ}\) ) 

= ( \(\sqrt {(sin²θ + cos²θ)/sin²θ .cos²θ }\) ) . ( \(\frac{ sinθ ( 1 + cosθ)}{ cosθ  + cos²θ}\) )

= \(\frac{1}{sinθ.cosθ}\) . ( \(\frac{ sinθ ( 1 + cosθ)}{ cosθ(1  + cosθ)}\) )

= \(\frac{1}{sinθ.cosθ}\) . ( \(\frac{ sinθ}{ cosθ}\) )

= \(\frac{1}{cos²θ}\)

= sec²θ