Find a simplified expression for cos(tan^(-1)(x/5))
To find a simplified expression for cos(tan^(-1)(x/5)), we can make use of the trigonometric identity:
cos(tan^(-1)(y)) = 1 / sqrt(1 + y^2)
In our case, y = x/5. Therefore, we can rewrite the expression as:
cos(tan^(-1)(x/5)) = 1 / sqrt(1 + (x/5)^2)
Now, let's simplify this expression further. First, let's square the denominator:
(1 + (x/5)^2) = (1 + x^2/25)
Now, we can take the square root:
sqrt(1 + x^2/25) = sqrt((25 + x^2) / 25)
Finally, we can substitute this back into our original expression:
cos(tan^(-1)(x/5)) = 1 / sqrt(1 + (x/5)^2) = 1 / sqrt((25 + x^2) / 25)
Hence, a simplified expression for cos(tan^(-1)(x/5)) is 1 / sqrt((25 + x^2) / 25).