Verify the identity:
arctanx + arctan(1/x)=∏/2
using calculus theory.
(Hint: Differentiate the left hand side of the identity)
Actually, since tan(pi/2-x) = cot(x)
and cot = 1/tan it falls right out.
Or, using the sum of tangents formula,
tan(arctanx + arctan(1/x))
= [tan(arctan(x)) + tan(arctan(1/x))][1 - tan(arctan(x))*tan(arctan(1/x))]
= [x + 1/x]/[1 - x*1/x] = (x + 1/x)/0 = oo
tan pi/2 = oo
To verify the given identity using calculus, we can differentiate both sides of the equation and show that they are equal.
Let's start by differentiating the left-hand side of the identity.
Using the property that the derivative of the inverse tangent function is given by:
d/dx(arctan(x)) = 1 / (1 + x^2),
we can differentiate arctan(x) with respect to x to obtain:
d/dx(arctan(x)) = 1 / (1 + x^2).
Next, we need to differentiate arctan(1/x) with respect to x. Since the argument of arctan is 1/x, we can use the chain rule of differentiation.
Let u = 1/x. Then, du/dx = -1/x^2.
Using the chain rule, we have:
d/dx(arctan(1/x)) = d/dx(arctan(u)) = (1 / (1 + u^2)) * du/dx
= (1 / (1 + (1/x)^2)) * (-1/x^2)
= (-1 / (1 + 1/x^2 * x^2))
= (-1 / (1 + 1))
= -1/2.
Now, let's combine the results we obtained and calculate the derivative of the left-hand side of the identity:
d/dx(arctan(x) + arctan(1/x)) = d/dx(arctan(x)) + d/dx(arctan(1/x))
= 1 / (1 + x^2) + (-1/2)
= (2 - (1 + x^2)) / (2(1 + x^2))
= (1 - x^2) / (2(1 + x^2))
= (1 - x^2) / (2 + 2x^2).
Now, we need to evaluate this derivative and check if it equals zero for every value of x. If it does, then the left-hand side of the identity is a constant, and we can equate it to the constant value on the right-hand side, which is π/2.
Setting the derivative equal to zero and solving for x:
(1 - x^2) / (2 + 2x^2) = 0.
Since the numerator (1 - x^2) is always positive (assuming x is real), the only way for the expression to equal zero is when the denominator (2 + 2x^2) becomes zero.
2 + 2x^2 = 0
Simplifying:
x^2 = -1.
This equation has no real solutions. Therefore, the derivative of the left-hand side of the identity is never zero. Hence, the left-hand side of the identity does not simplify to a constant, and it cannot be equated to π/2.
Therefore, we can conclude that the given identity is not true.