Answered

Which expression is equivalent to [tex] \tan \left(\frac{3 \pi}{4}-2 x\right) [/tex]?

A. [tex] \frac{-1-\tan (2 x)}{1-(-1)(\tan (2 x))} [/tex]
B. [tex] \frac{-1-\tan (2 x)}{1+(-1)(\tan (2 x))} [/tex]
C. [tex] \frac{1-\tan (2 x)}{1+(1)(\tan (2 x))} [/tex]
D. [tex] \frac{1+\tan (2 x)}{1-(1)(\tan (2 x))} [/tex]



Answer :

GinnyAnswer
To find which expression is equivalent to \(\tan \left(\frac{3 \pi}{4} - 2x\right)\), we can use the tangent subtraction formula:

[tex]\[ \tan(a - b) = \frac{\tan(a) - \tan(b)}{1 + \tan(a) \tan(b)} \][/tex]

Here, let \(a = \frac{3 \pi}{4}\) and \(b = 2x\).

First, we need to determine \(\tan\left(\frac{3 \pi}{4}\right)\). Knowing the tangent values of common angles, we have:

[tex]\[ \tan\left(\frac{3 \pi}{4}\right) = \tan\left(135^\circ\right) = -1 \][/tex]

Using the tangent subtraction formula:

[tex]\[ \tan\left(\frac{3 \pi}{4} - 2x\right) = \frac{\tan\left(\frac{3 \pi}{4}\right) - \tan(2x)}{1 + \tan\left(\frac{3 \pi}{4}\right) \tan(2x)} \][/tex]

Substitute \(\tan\left(\frac{3 \pi}{4}\right) = -1\):

[tex]\[ \tan\left(\frac{3 \pi}{4} - 2x\right) = \frac{-1 - \tan(2x)}{1 + (-1) \cdot \tan(2x)} \][/tex]

Simplify the denominator:

[tex]\[ \tan\left(\frac{3 \pi}{4} - 2x\right) = \frac{-1 - \tan(2x)}{1 - \tan(2x)} \][/tex]

Thus, the expression equivalent to \(\tan\left(\frac{3 \pi}{4} - 2x\right)\) is:

[tex]\[ \frac{-1 - \tan(2x)}{1 - \tan(2x)} \][/tex]

So, the correct choice is the first option:
[tex]\[ \boxed{\frac{-1 - \tan(2x)}{1 - \tan(2x)}} \][/tex]

Other Questions