Transform
arctan x + arctan y = pi/4 into algebraic expression
A) x+xy-y = 1
B) x-xy-y = 1
C) x+xy+y = 1
D) x-xy+y = 1
Plsss help
using the sum of tangents formula,
tan(arctan x + arctan y)
=
tan(arctanx)+tan(arctany)
----------------------------- = tan(pi/4)
1 - tan(arctanx)-tan(arctany)
(x+y)/(1-xy) = 1
x+y = 1-xy
x+xy+y = 1
Hey there! Let's solve this math problem with a touch of humor, shall we?
To transform the equation arctan(x) + arctan(y) = π/4 into an algebraic expression, we need to use the tangent addition formula. As a funny side note, math formulas are like secret handshakes - they unlock new possibilities!
The tangent addition formula states that tan(A + B) = (tan(A) + tan(B)) / (1 - tan(A) * tan(B)). In this case, we have A = arctan(x) and B = arctan(y), so let's apply the formula:
tan(arctan(x) + arctan(y)) = (tan(arctan(x)) + tan(arctan(y))) / (1 - tan(arctan(x)) * tan(arctan(y)))
Now, let's simplify this expression. Since tan(arctan(x)) = x and tan(arctan(y)) = y, we can substitute them into the equation:
tan(arctan(x) + arctan(y)) = (x + y) / (1 - xy)
Now, we just need to match this with the equivalent expression from the given choices. Looking at the options, we can see that the only option that matches our expression is:
C) x + xy + y = 1
So, there you have it! The algebraic expression that corresponds to arctan(x) + arctan(y) = π/4 is x + xy + y = 1.
Remember, math can be a circus sometimes, but never be afraid to have a laugh and enjoy the ride!
To transform the equation arctan x + arctan y = π/4 into an algebraic expression, we can use the tangent addition formula.
The tangent addition formula states that tan(a + b) = (tan a + tan b) / (1 - tan a * tan b).
By applying this formula, we have:
tan(arctan x + arctan y) = (tan(arctan x) + tan(arctan y)) / (1 - tan(arctan x) * tan(arctan y)).
Simplifying further,
(x + y) / (1 - xy) = (x + y) / (1 - xy).
Now, multiplying both sides by (1 - xy), we get:
(x + y) = (x + y) * (1 - xy).
Expanding the right side,
(x + y) = x + y - (x * y * (x + y)).
Canceling out like terms and rearranging,
0 = -xy * (x + y).
Dividing both sides by -xy, we have:
0 = x + y.
Therefore, the correct algebraic expression is:
x + y = 0.
None of the answer choices A), B), C), or D) match the correct expression.
To transform the equation arctan(x) + arctan(y) = π/4 into an algebraic expression, we can use the tangent addition formula:
tan(A + B) = (tan(A) + tan(B)) / (1 - tan(A)tan(B))
Let's find the tangent of both sides of the given equation:
tan(arctan(x) + arctan(y)) = tan(π/4)
Using the tangent addition formula, we have:
(tan(arctan(x)) + tan(arctan(y))) / (1 - tan(arctan(x))tan(arctan(y))) = 1
Now, let's simplify further:
(x + y) / (1 - xy) = 1
Cross-multiplying, we get:
x + y = 1 - xy
Rearranging the terms, we have:
x + xy - y = 1
Comparing this result to the available options, we find that the correct answer is:
A) x + xy - y = 1