In Exercises 9-50, verify the identity $\frac{1}{\tan x}+\frac{1}{\cot x}=\tan x+\cot x$

Step by Step Solution: Step 1. Use the reciprocal identities $\tan x=\frac{1}{\cot x}$ and $\cot x=\frac{1}{\tan x}$ to rewrite the left-hand side: Step 2. $\frac{1}{\tan x}+\frac{1}{\cot x}=\frac{\cot x}{\cot x\tan x}+\frac{\tan x}{\cot x\tan x}$ Step 3. Simplify the fractions on the right-hand side of the equation: Step 4. $\frac{\cot x}{\cot x\tan x}+\frac{\tan x}{\cot x\tan x}=\frac{{\cot }^{2}x+{\tan }^{2}x}{\cot x\tan x}$ Step 5. Use the Pythagorean identity ${\cot }^{2}x+1={\csc }^{2}x$ and ${\tan }^{2}x+1={\sec }^{2}x$ to simplify the numerator: Step 6. ${\cot }^{2}x+{\tan }^{2}x=\frac{{\cos }^{2}x}{{\sin }^{2}x}+\frac{{\sin }^{2}x}{{\cos }^{2}x}=\frac{{\cos }^{4}x+{\sin }^{4}x}{{\cos }^{2}x{\sin }^{2}x}$ Step 7. Combine the simplified fractions from steps 4 and 6: Step 8. $\frac{{\cot }^{2}x+{\tan }^{2}x}{\cot x\tan x}=\frac{{\cos }^{4}x+{\sin }^{4}x}{{\cos }^{2}x{\sin }^{2}x\cot x\tan x}$ Step 9. Use the identity ${\sin }^{2}x=1-{\cos }^{2}x$ to simplify the numerator on the right-hand side of the equation: Step 10. ${\cos }^{4}x+{\sin }^{4}x=({\cos }^{2}x{)}^2+(1-{\cos }^{2}x{)}^2=2{\cos }^{4}x-2{\cos }^{2}x+1$ Step 11. Substitute the simplified numerator from step 10 into the equation from step 8: Step 12. $\frac{{\cos }^{4}x+{\sin }^{4}x}{{\cos }^{2}x{\sin }^{2}x\cot x\tan x}=\frac{2{\cos }^{4}x-2{\cos }^{2}x+1}{{\cos }^{2}x{\sin }^{2}x\cot x\tan x}$ Step 13. Use the identity $\cot x\tan x=\frac{1}{{\sin }^{2}x}$ to simplify the denominator: Step 14. $\frac{2{\cos }^{4}x-2{\cos }^{2}x+1}{{\cos }^{2}x{\sin }^{2}x\cot x\tan x}=\frac{2{\cos }^{4}x-2{\cos }^{2}x+1}{{\cos }^{2}x}$ Step 15. Use the identity ${\cos }^{2}x=1-{\sin }^{2}x$ to simplify the numerator: Step 16. $2{\cos }^{4}x-2{\cos }^{2}x+1=2(1-{\sin }^{2}x{)}^2-2(1-{\sin }^{2}x)+1$ Step 17. Simplify the numerator using algebra: Step 18. $2(1-{\sin }^{2}x{)}^2-2(1-{\sin }^{2}x)+1=2(1-2{\sin }^{2}x+{\sin }^{4}x)-2+1$ Step 19. $2(1-2{\sin }^{2}x+{\sin }^{4}x)-2+1=2{\sin }^{4}x-4{\sin }^{2}x+3$ Step 20. Substitute the simplified numerator from step 19 into the equation from step 14: Step 21. $\frac{2{\cos }^{4}x-2{\cos }^{2}x+1}{{\cos }^{2}x}=\frac{2{\sin }^{4}x-4{\sin }^{2}x+3}{{\sin }^{2}x}$ Step 22. Use the identity ${\csc }^{2}x=1+{\cot }^{2}x$ to simplify the denominator of the right-hand side of the equation: Step 23. $\frac{2{\sin }^{4}x-4{\sin }^{2}x+3}{{\sin }^{2}x}=2(1+{\cot }^{2}x)-4{\csc }^{2}x+3$ Step 24. Use the identity $\cot x=\frac{1}{\tan x}$ and substitute it in the equation: Step 25. $\frac{2{\cos }^{4}x-2{\cos }^{2}x+1}{{\cos }^{2}x}=2+\frac{2}{{\tan }^{2}x}-4(1+{\csc }^{2}x)+3$ Step 26. Use the identity $\tan x+\cot x=\frac{\sin x}{\cos x}+\frac{\cos x}{\sin x}$ to simplify the right-hand side of the equation: Step 27. $2+\frac{2}{{\tan }^{2}x}-4(1+{\csc }^{2}x)+3=-3+\frac{2{\cos }^{2}x}{{\sin }^{2}x}+\frac{2{\sin }^{2}x}{{\cos }^{2}x}=-3+2(\tan x+\cot x)$ Final Answer: -3 + 2(\tan x + \cot x)