tan200(cot10-tan10)
To solve this expression, we need to use trigonometric identities and simplification techniques. Here's how you can approach it step by step:
Step 1: Simplify the expression inside the parentheses.
Inside the parentheses, we have cot(10) - tan(10). We know that cot(θ) is the reciprocal of tan(θ), so cot(10) is equal to 1/tan(10). Therefore, we can rewrite the expression as tan(200) * (1/tan(10) - tan(10)).
Step 2: Simplify the expression further.
Using the formula tan(a - b) = (tan(a) - tan(b)) / (1 + tan(a)tan(b)), we can rewrite the expression as tan(200) * (1/tan(10) - tan(10)) = tan(200) * (tan(10) - 1/tan(10)).
Step 3: Apply the tangent addition formula.
The tangent addition formula states that tan(a ± b) = (tan(a) ± tan(b)) / (1 ∓ tan(a)tan(b)). Applying the formula, we can rewrite tan(200) * (tan(10) - 1/tan(10)) as tan(200 + 10) = tan(210).
Step 4: Calculate the value.
Use a calculator or a trigonometric table to find tan(210). The exact value of tan(210) is -√3 or approximately -1.732.
Therefore, tan(200)(cot(10) - tan(10)) is equal to -√3 or approximately -1.732.