Question

When simplified, (tan(40°)-tan(10°))/(1+tan(40°)-tan(10°)) is equivalent to which expression?

A. tan(–50°)
B. tan(–30°)
C. tan(30°)
D. tan(50°)

Answer 1

The simplified expression
`\((\tan(40^\circ)-\tan(10^\circ))/(1+\tan(40^\circ)-\tan(10^\circ))\)` is equivalent to tan(30°)(C).

Step-by-step explanation:

To simplify the given expression
`\((\tan(40^\circ)-\tan(10^\circ))/(1+\tan(40^\circ)-\tan(10^\circ))\),` let's first use the tangent subtraction formula:
`\(\tan(A) - \tan(B) = (\tan(A) - \tan(B))/(1 + \tan(A) \cdot \tan(B))\)`. Applying this formula, we get:

`\((\tan(40^\circ)-\tan(10^\circ))/(1+\tan(40^\circ)-\tan(10^\circ)) = ((\tan(40^\circ)-\tan(10^\circ))/(1 + \tan(40^\circ) \cdot \tan(10^\circ)))/((1+\tan(40^\circ)-\tan(10^\circ))/(1 + \tan(40^\circ) \cdot \tan(10^\circ)))\).`

Simplify the numerator and denominator:

Numerator:
`\(\tan(40^\circ)-\tan(10^\circ) = \tan(30^\circ)\)` using the tangent subtraction formula \(\tan(A) - \tan(B) = \tan(A-B)\).

Denominator:
`\(1+\tan(40^\circ)-\tan(10^\circ) = 1+\tan(30^\circ) = 1 + (1)/(√(3))\).`

Therefore, the simplified expression becomes
`\((\tan(30^\circ))/(1 + (1)/(√(3)))\).` Rationalizing the denominator by multiplying the numerator and denominator by
`\(√(3)\), we get \(\tan(30^\circ) * √(3) = √(3)\).`

Hence, the simplified expression is
`\(\tan(30^\circ)\)`, which is equivalent to
`\(√(3)\)` in exact form. so the correct option is tan(30°)(C).