College

If [tex]\cot(x/2) = \frac{4}{3}[/tex], then the value of [tex]\cos x[/tex] is:

A. [tex]\frac{2}{5}[/tex]
B. [tex]\frac{3}{25}[/tex]
C. [tex]\frac{7}{25}[/tex]
D. [tex]\frac{12}{25}[/tex]

Answer :

Final answer:

To find the value of cos(x) given cot(x/2), use the identity cos(x) = sqrt[(1+cos(2x))/2]. Simplify the equation cot(x/2) = sqrt[(1+cos(x))/(1-cos(x))], solve for cos(x) to get 7/25.


Explanation:

To find the value of cos(x), we need to use the identity cos(x) = ±√[(1+cos(2x))/2].

Since cot(x/2) = 4/3, we can use the identity cot(x/2) = ±√[(1+cos(x))/ (1-cos(x))].

  1. Using the given value of cot(x/2) = 4/3, we can solve for cos(x). First, square both sides of the equation to get (cot(x/2))^2 = (4/3)^2.
  2. Then, simplify to get (1+cos(x))/(1-cos(x)) = 16/9.
  3. Cross-multiply and simplify to obtain 9 + 9cos(x) = 16 - 16cos(x).
  4. Combine like terms and rearrange the equation to get 25cos(x) = 7.
  5. Finally, divide by 25 to find the value of cos(x) = 7/25.

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