cot^-1(2tan(cos^-18/17))+tan^-1(2tan(sin^-18/17))=tan^-1300/161

the cos^-18/17 and the sin^-18/17 in your expression don't make sense

did you mean cos^-1 (18/17) ?
in that case, that would be undefined, since no angle exists such that cosØ = 18/17
same thing would be true for the sine part

To simplify the given expression, we can start by using trigonometric identities and properties. We'll break it down step by step.

1. Let's assign some variables for convenience.
- Let alpha = cos^(-1)(18/17)
- Let beta = sin^(-1)(18/17)

2. Applying the identity: tan(x) = sin(x)/cos(x), we can rewrite the expression as:
- cot^(-1)(2 * (sin(alpha)/cos(alpha))) + tan^(-1)(2 * (sin(beta)/cos(beta))) = tan^(-1300/161)

3. Using the identity: cot^(-1)(x) = tan^(-1)(1/x), we can rewrite the first part of the expression as:
- tan^(-1)(1 / (2 * (sin(alpha)/cos(alpha))))

4. Simplify the expression inside the arctan:
- tan^(-1)((cos(alpha)/sin(alpha)) / (2 * (sin(alpha)/cos(alpha))))
- tan^(-1)(cos(alpha) / (2 * sin(alpha)))

5. Using the identity: tan^(-1)(x/y) = tan^(-1)(x) - tan^(-1)(y), we can rewrite the expression as:
- tan^(-1)(cos(alpha)) - tan^(-1)(2 * sin(alpha))

6. Now, let's simplify the second part of the expression:
- tan^(-1)(2 * (sin(beta)/cos(beta)))
- tan^(-1)((cos(beta)/sin(beta)) / (2 * (sin(beta)/cos(beta))))
- tan^(-1)(cos(beta) / (2 * sin(beta)))

7. By applying the same identity, we can rewrite it as:
- tan^(-1)(cos(beta)) - tan^(-1)(2 * sin(beta))

8. Now the equation becomes:
- (tan^(-1)(cos(alpha)) - tan^(-1)(2 * sin(alpha))) + (tan^(-1)(cos(beta)) - tan^(-1)(2 * sin(beta))) = tan^(-1300/161)

9. By rearranging the terms, we have:
- [tan^(-1)(cos(alpha)) + tan^(-1)(cos(beta))] - [tan^(-1)(2 * sin(alpha)) + tan^(-1)(2 * sin(beta))] = tan^(-1300/161)

At this point, the equation cannot be further simplified algebraically. To determine whether the equation holds true, you would need to use a numerical approach, such as approximation methods or a calculator.