Answer: LHS simplifies to RHS, as shown below.
Divide numerator and denominator of the LHS by \(\cos\theta\):
\[\text{LHS}=\frac{\tan\theta+\sec\theta-1}{\tan\theta-\sec\theta+1}\]
In the numerator write \(1=\sec^2\theta-\tan^2\theta\):
\[\tan\theta+\sec\theta-(\sec\theta-\tan\theta)(\sec\theta+\tan\theta)\]
\[=(\sec\theta+\tan\theta)(1-\sec\theta+\tan\theta)\]
The bracket \((1-\sec\theta+\tan\theta)\) equals the denominator, so it cancels:
\[\text{LHS}=\sec\theta+\tan\theta\]
And \(\sec\theta+\tan\theta=\dfrac{\sec^2\theta-\tan^2\theta}{\sec\theta-\tan\theta}=\dfrac{1}{\sec\theta-\tan\theta}=\text{RHS}\).