MHB Write a trigonometric expression as an algebraic expression

AI Thread Summary
The discussion revolves around converting the trigonometric expression cos(arccos x + arcsin x) into an algebraic form. The key identity used is arccos(x) + arcsin(x) = π/2, which simplifies the expression. By applying the angle-sum identity for cosine, the expression can be rewritten as a product of cosine and sine functions. The final result of the expression is determined to be zero. The participants express clarity and understanding as the problem is worked through collaboratively.
Taryn1
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This problem probably should be easy, but I don't remember learning the basic way to do these problems: Write the trigonometric expression as an algebraic expression:

cos(arccos x + arcsin x)

The answer is zero, but I don't know how to get there...
 
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Recall the identity:

$$\arccos(x)+\arcsin(x)=\frac{\pi}{2}$$

However, I suspect you are to use the angle-sum identity for cosine to write:

$$\cos\left(\arccos(x)+\arcsin(x)\right)=\cos\left(\arccos(x)\right)\cos\left(\arcsin(x)\right)-\sin\left(\arccos(x)\right)\sin\left(\arcsin(x)\right)$$

Can you continue?
 
Ohhhh, I think I get it now. Thanks!
 
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