Final answer:
The equation of the tangent line to the curve y = cos(x) − sin(x) at the point (π, −1) is y = x - π - 1. To find this, we calculate the derivative to get the slope of the tangent line at π and then use the point-slope form to write the equation.
Step-by-step explanation:
To find the equation of the tangent line to the curve y = cos(x) − sin(x) at the point (π, −1), we need to first calculate the derivative of the given function which will give us the slope of the tangent line at any point x.
The derivative of y with respect to x is y' = -sin(x) - cos(x). We evaluate the derivative at x = π to get the slope at the point of tangency: y'(π) = -sin(π) - cos(π) = 0 - (-1) = 1. Therefore, the slope of the tangent line at (π, −1) is 1.
Now, we use the point-slope form of the equation for a line: y - y1 = m(x - x1), where m is the slope and (x1, y1) is the point of tangency. Plugging in the slope and the coordinates of the given point, we get the equation of the tangent line: y - (-1) = 1(x - π), which simplifies to y = x - π - 1.