Final answer:
To find the value of cos A, rearrange the given equation and solve the resulting quadratic equation. The only valid solution for cos A is 1.
Step-by-step explanation:
To find the value of cos A, we need to rearrange the given equation tan A + cos A = 2. Let's subtract cos A from both sides of the equation to isolate the tan A term: tan A = 2 - cos A. Now we can use the identity tan ^2 A + 1 = sec ^2 A to rewrite the equation: (2 - cos A) ^2 + 1 = sec ^2 A. Expanding and simplifying: 4 - 4cos A + cos ^2 A + 1 = sec ^2 A. Rearranging and simplifying further: cos ^2 A + 5cos A - 3 = 0.
Solving this quadratic equation for cos A, we can factor it as (cos A - 1)(cos A + 3) = 0. This gives us two possible values for cos A: cos A = 1 or cos A = -3. However, the range of the cosine function is -1 to 1, so the only valid solution is cos A = 1.