how do i evaluate this?

1+tan 200(degrees) tan 80(degrees)/tan200(degrees)-tan 80(degrees)

I am sure you meant:

(1+tan 200° tan 80°)/ (tan200°-tan 80°)
--- those brackets are critical

recall:
tan(A-B)
= (tanA - TanB)/( 1 + tanAtanB)
= 1/[ (1+tanAtanB)/(tanA - tanB) ]

tan(200-80) = tan 120 = - tan60° = -√3
= (tan 200 - tan80)/(1 + tan200tan80)
= -1/√3

Recall the sum of angles for tan:

tan(A-B) = (tanA-tanB)/(1+tanAtanB)

You have the reciprocal of that, so what you have is

cot(200-80) = cot(120) = -1/√3

To evaluate the expression 1 + tan(200°) tan(80°) / tan(200°) - tan(80°), you can follow these steps:

Step 1: Convert the given angles from degrees to radians.
The conversion formula is: radians = degrees * (π/180)

Converting the given angles:
200° to radians: 200 * (π/180) = 10π/9
80° to radians: 80 * (π/180) = 4π/9

Step 2: Substitute the converted values into the expression.

1 + tan(10π/9) tan(4π/9) / tan(10π/9) - tan(4π/9)

Step 3: Evaluate the expressions involving tangent.

tan(a + b) = (tan a + tan b) / (1 - tan a tan b)

Applying this formula to the expression, let a = 10π/9 and b = 4π/9:

1 + (tan 10π/9 + tan 4π/9) / (1 - tan 10π/9 tan 4π/9) / (tan 10π/9 - tan 4π/9)

Step 4: Simplify the expression.

Let's simplify the numerator first:

tan(10π/9) + tan(4π/9) = (sin(10π/9) / cos(10π/9)) + (sin(4π/9) / cos(4π/9))

Using the identities: sin(a + b) = sin a cos b + cos a sin b and cos(a + b) = cos a cos b - sin a sin b

sin(10π/9) = sin(π/9 + π) = sin(π/9) cos(π) + cos(π/9) sin(π) = sin(π/9)
cos(10π/9) = cos(π/9 + π) = cos(π/9) cos(π) - sin(π/9) sin(π) = -cos(π/9)

sin(4π/9) = sin(π/9 + π/4) = sin(π/9) cos(π/4) + cos(π/9) sin(π/4) = (1/√2) sin(π/9) + (1/√2) cos(π/9)
cos(4π/9) = cos(π/9 + π/4) = cos(π/9) cos(π/4) - sin(π/9) sin(π/4) = (1/√2) cos(π/9) - (1/√2) sin(π/9)

Substituting these values:

= [(sin(π/9) / -cos(π/9)) + ((1/√2) sin(π/9) + (1/√2) cos(π/9))] / (1 - (-cos(π/9)) ((1/√2) cos(π/9) - (1/√2) sin(π/9)))

Simplifying further:

= [sin(π/9) - (1/√2) sin(π/9) - (1/√2) cos(π/9)] / (1 + (1/√2) cos(π/9) - (1/√2) sin(π/9) + cos^2(π/9) + sin^2(π/9))

= [sin(π/9) - (1/√2) (sin(π/9) + cos(π/9))] / (1 + (1/√2) (cos(π/9) - sin(π/9)) + 1)

= [sin(π/9) - (1/√2) sin(π/4)] / (2 + (1/√2) (cos(π/9) - sin(π/9)))

Simplifying the denominator:

= [sin(π/9) - (1/√2) sin(π/4)] / (2 + (1/√2) cos(π/9) - (1/√2) sin(π/9))

Now, let's evaluate the denominator:

cos(π/9) = cos(π/3 - π/9) = cos(π/3) cos(π/9) + sin(π/3) sin(π/9) = (1/2) cos(π/9) + (√3/2) sin(π/9)

Substituting into the denominator:

= [sin(π/9) - (1/√2) sin(π/4)] / (2 + (1/√2) ((1/2) cos(π/9) + (√3/2) sin(π/9)) - (1/√2) sin(π/9))

= [sin(π/9) - (1/√2) sin(π/4)] / (2 + (1/2√2) cos(π/9) + (√3/2√2) sin(π/9) - (1/√2) sin(π/9))

= [sin(π/9) - (1/√2) sin(π/4)] / (2 - (√3/2√2) sin(π/9) + (1/2√2) cos(π/9))

Now, let's return to the original expression:

1 + tan(10π/9) tan(4π/9) / tan(10π/9) - tan(4π/9) = 1 + [(sin(π/9) - (1/√2) sin(π/4)) / (2 - (√3/2√2) sin(π/9) + (1/2√2) cos(π/9))] / (2 - (√3/2√2) sin(π/9) + (1/2√2) cos(π/9))

Note: Unfortunately, it is not possible to simplify this expression further without any specific values for π. If you have the exact value for π, you can substitute it into the expression and then use a calculator to evaluate it numerically.

To evaluate the given expression, we can use the trigonometric identities to simplify it step by step. Here's the process:

Step 1: Convert degrees to radians
First, convert the given angles from degrees to radians. Use the following conversion factor: π radians = 180 degrees
- 200 degrees = 200π/180 radians ≈ (10π/9) radians
- 80 degrees = 80π/180 radians ≈ (4π/9) radians

Step 2: Simplify the expression
Substitute the values into the expression:
1 + tan((10π/9)) * tan((4π/9)) / (tan((10π/9)) - tan((4π/9)))

Step 3: Apply trigonometric identities
Use the identity tan(A + B) = (tan A + tan B) / (1 - tan A * tan B) to rewrite the expression:
1 + (tan((10π/9)) + tan((4π/9))) / (1 - tan((10π/9)) * tan((4π/9))) / (tan((10π/9)) - tan((4π/9)))

Step 4: Simplify further
To simplify this expression, substitute the values of tan((10π/9)) and tan((4π/9)) from the unit circle or use a calculator.

Finally, you can solve the expression by simplifying any further mathematical operations like addition, subtraction, or division.

Note: It is recommended to use a scientific calculator or an online trigonometric calculator to evaluate the expression accurately, especially when dealing with trigonometric functions involving angles in radians.