Solve without the use of a calculator:

sin20(tan10+cot10)

I know it should equal two but I don't know how to get there

sin20(tan10+cot10)

= sin20(sin10/cos10 + cos10/sin10)
= 2sin10cos10(sin^2 10 + cos^2 10)/(sin10cos10)
= 2(sin10cos10)/(sin10cos10 ( 1)
= 2

To solve the expression without a calculator, we can use trigonometric identities and simplification techniques. Let's break down the problem step by step:

1. Starting with the expression sin(20°(tan(10°) + cot(10°))), we can simplify the inside part first.

2. The tangent (tan) and cotangent (cot) functions are reciprocal of each other. Therefore, tan(10°) is equal to 1/cot(10°) and cot(10°) is equal to 1/tan(10°). So, tan(10°) + cot(10°) can be rewritten as 1/tan(10°) + 1/cot(10°).

3. To add these fractions, we need a common denominator. Since tan(10°) and cot(10°) have a common denominator of 1, the common denominator for the sum is also 1.

4. Adding the fractions, we get (1 + 1)/(tan(10°)*cot(10°)), which simplifies to 2/(tan(10°)*cot(10°)).

5. Now, we can simplify further. The product of the tangent and cotangent of the same angle is always equal to 1. So, tan(10°) * cot(10°) = 1, and our expression becomes 2/1, which simplifies to just 2.

Therefore, the value of sin(20°(tan(10°) + cot(10°))) is equal to 2.