Answer:
D) None
Explanation:
To solve the given expression, let's start by using the trigonometric identity:
tan(A + B) = (tan A + tan B) / (1 - tan A * tan B)
We can rewrite the expression as follows:
tan 20 + tan 40 + √3 × tan 20 × tan 40
Using the identity, we have:
tan(20 + 40) = (tan 20 + tan 40) / (1 - tan 20 * tan 40)
tan 60 = √3
Now let's substitute tan(60) with √3:
(√3) / (1 - tan 20 * tan 40) + √3 × tan 20 × tan 40
We need to find the value of tan 20 * tan 40. To do that, let's use another trigonometric identity:
tan(A - B) = (tan A - tan B) / (1 + tan A * tan B)
tan(40 - 20) = (tan 40 - tan 20) / (1 + tan 40 * tan 20)
tan 20 = (tan 40 - tan 20) / (1 + tan 40 * tan 20)
Rearranging the equation, we get:
tan 20 + tan 20 * tan 40 = tan 40
Now, let's substitute tan 40 with (tan 20 + tan 20 * tan 40):
(√3) / (1 - tan 20 * tan 40) + √3 × tan 20 × tan 40
= (√3) / (1 - (tan 20 + tan 20 * tan 40)) + √3 × (tan 20 + tan 20 * tan 40) × tan 20 × tan 40
= (√3) / (1 - (tan 20 + tan 20 * (tan 20 + tan 20 * tan 40))) + √3 × (tan 20 + tan 20 * (tan 20 + tan 20 * tan 40)) × tan 20 × tan 40
= (√3) / (1 - (tan 20 + tan 20 * (tan 20 + tan 20 * (tan 20 + tan 20 * tan 40)))) + √3 × (tan 20 + tan 20 * (tan 20 + tan 20 * (tan 20 + tan 20 * tan 40))) × tan 20 × tan 40
By substituting recursively, we can continue this process indefinitely. Therefore, the expression doesn't have a finite solution. The answer is D) None.