cos(arcsin(x)+arctan(x))
Let arcsin(x) = A
let arctan(x) = B
then cos(arcsin(x) + arctan(x))
= cos(A+B)
= cosAcosB - sinAsinB **
from arcsin(x) = A
sinA = x/1
cosA = √(1-x^2) , using Pythagoras
from arctan(x) = B
tanB = x/1 ,
then sinB = x/√(x^2 + 1)
cosB = 1/√(x^2 + 1)
back into **
cosAcosB - sinAsinB
= √(1 - x^2) (1/√(x^2 + 1) - (x/1)(x/√(x^2 + 1)
= ( √(1 - x^2) - x^2 )/(√(x^2 + 1) )
check my typing, should have written it out first.
Well, buckle up math adventurer, because we're about to embark on a journey filled with trigonometric shenanigans!
Let's start by breaking down this expression step by step:
1. First, we have arcsin(x). This represents the inverse sine function, which gives us the angle whose sine is x. It's like trying to find the person who ate all the pi-shaped cookies in the middle of the night... sneaky, sneaky!
2. Next up, we have arctan(x). This represents the inverse tangent function, which gives us the angle whose tangent is x. Think of it as trying to find the angle at which two monkeys are hanging on a tree branch, causing all sorts of monkey business!
3. Now, we need to add these two angles together, like mixing peanut butter and jelly to create the ultimate trigonometric sandwich!
4. Finally, we have cos(...) which represents the cosine function. It's like the chill cousin of sine and tangent, always hanging out in the shadow, waiting for its moment to shine!
So, when you sum arcsin(x) and arctan(x) together, and then take the cosine of that sum, you get the answer to your question. Unfortunately, this expression doesn't simplify nicely into a neat formula or a punchline. It's a mathematical adventure that's more delightful than a clown on a unicycle!
To find the value of cos(arcsin(x) + arctan(x)), we can use trigonometric identities. First, let's simplify the expression using the appropriate identity:
cos(arcsin(x) + arctan(x)) = cos(arcsin(x)) * cos(arctan(x)) - sin(arcsin(x)) * sin(arctan(x))
Using the identity sin^2θ + cos^2θ = 1, we can simplify further:
cos(arcsin(x) + arctan(x)) = cos(arcsin(x)) * cos(arctan(x)) - sin(arcsin(x)) * sin(arctan(x))
= √(1 - sin^2(arcsin(x))) * cos(arctan(x)) - x * sin(arctan(x))
Simplifying inside the square root:
= √(1 - x^2) * cos(arctan(x)) - x * sin(arctan(x))
Next, we substitute the values of cos(arctan(x)) and sin(arctan(x)) using the right triangle definition of tangent:
cos(arctan(x)) = 1 / √(1 + x^2)
sin(arctan(x)) = x / √(1 + x^2)
Substituting these values into the expression:
= √(1 - x^2) * (1 / √(1 + x^2)) - x * (x / √(1 + x^2))
= √(1 - x^2) / √(1 + x^2) - x^2 / √(1 + x^2)
= (√(1 - x^2) - x^2) / √(1 + x^2)
So, the value of cos(arcsin(x) + arctan(x)) is (√(1 - x^2) - x^2) / √(1 + x^2).
To find the value of cos(arcsin(x) + arctan(x)), we can use trigonometric identities and the properties of inverse trigonometric functions.
Let's begin by simplifying the expression inside the cosine function.
Step 1: Start with the expression cos(arcsin(x) + arctan(x))
Step 2: Use the identity sin(arcsin(x)) = x. This simplifies the arcsin(x) term: cos(arcsin(x)) + arctan(x)
Now, let's handle each term separately.
For cos(arcsin(x)), we need to use the identity cos(arcsin(x)) = sqrt(1 - x^2).
Step 3: Replace cos(arcsin(x)) with sqrt(1 - x^2): sqrt(1 - x^2) + arctan(x)
Lastly, let's simplify the arctan(x) term.
Step 4: Use the identity arctan(x) = arctan(x) to keep it as is: sqrt(1 - x^2) + arctan(x)
Therefore, the simplified expression is sqrt(1 - x^2) + arctan(x).
Note: The simplified expression cannot be further simplified as it is a combination of the square root and arctan functions. It is also important to remember that the resulting expression of cos(arcsin(x) + arctan(x)) depends on the value of x.