1. **State the problem:** Simplify the expression $$\sec \Theta + \frac{\csc \Theta \tan \Theta}{\sec^2 \Theta}$$ and verify if it equals $$\cot \Theta \csc \Theta$$. 2. **Rewrite the expression in terms of sine and cosine:** $$\sec \Theta = \frac{1}{\cos \Theta}$$ $$\csc \Theta = \frac{1}{\sin \Theta}$$ $$\tan \Theta = \frac{\sin \Theta}{\cos \Theta}$$ $$\sec^2 \Theta = \frac{1}{\cos^2 \Theta}$$ So, $$\sec \Theta + \frac{\csc \Theta \tan \Theta}{\sec^2 \Theta} = \frac{1}{\cos \Theta} + \frac{\frac{1}{\sin \Theta} \cdot \frac{\sin \Theta}{\cos \Theta}}{\frac{1}{\cos^2 \Theta}}$$ 3. **Simplify inside the fraction:** The numerator inside the large fraction is $$\frac{1}{\sin \Theta} \times \frac{\sin \Theta}{\cos \Theta} = \frac{1}{\cos \Theta}$$ The denominator is $$\frac{1}{\cos^2 \Theta}$$ Thus, $$\frac{\frac{1}{\cos \Theta}}{\frac{1}{\cos^2 \Theta}} = \frac{1}{\cos \Theta} \times \frac{\cos^2 \Theta}{1} = \cos \Theta$$ 4. **Rewrite the full expression now:** $$\frac{1}{\cos \Theta} + \cos \Theta$$ 5. **Simplify and compare with the right side:** Right side is: $$\cot \Theta \csc \Theta = \frac{\cos \Theta}{\sin \Theta} \times \frac{1}{\sin \Theta} = \frac{\cos \Theta}{\sin^2 \Theta}$$ The left side simplified to: $$\frac{1}{\cos \Theta} + \cos \Theta$$ The right side is: $$\frac{\cos \Theta}{\sin^2 \Theta}$$ 6. **Check equality:** Since these two expressions are generally different, the original equality does not hold. **Final conclusion:** $$\sec \Theta + \frac{\csc \Theta \tan \Theta}{\sec^2 \Theta} \neq \cot \Theta \csc \Theta$$.