SOLUTION: I need help verifying this identity, factoring is confusing me. Shouldn't I begin by factoring the top on the left side?
cos 5x cos 3x + sin 5X sin 3x CSC X
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Question 132787: I need help verifying this identity, factoring is confusing me. Shouldn't I begin by factoring the top on the left side?
cos 5x cos 3x + sin 5X sin 3x CSC X
----------------------------- = -------
sin 3x - sin X 2
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Note: I'm only manipulating the left side. I'm showing the right just for comparison.
\cos \left(3x\right)+\sin \left(5x\right)\sin \left(3x\right)}{\sin \left(3x\right)-\sin \left(x\right)}=\frac{\csc\left(x\right)}{2})
Start with the given equation
}{\sin \left(3x\right)-\sin \left(x\right)}=\frac{\csc\left(x\right)}{2})
Condense the numerator by using the identity
}{\sin \left(3x\right)-\sin \left(x\right)}=\frac{\csc\left(x\right)}{2})
Combine like terms
}{\sin \left(3x\right)-\sin \left(x\right)}=\frac{\csc\left(x\right)}{2})
Use the identity =1-2\sin^2\left(x\right))
to replace the numerator
}{\sin \left(2x+x\right)-\sin \left(x\right)}=\frac{\csc\left(x\right)}{2})
Expand 3x to 2x+x
}{\sin\left(2x\right)\cos\left(x\right)+\cos\left(2x\right)\sin\left(x\right)-\sin \left(x\right)}=\frac{\csc\left(x\right)}{2})
Expand through this identity =\sin\left(x\right)\cos\left(y\right)+\cos\left(x\right)\sin\left(y\right))
}{2\sin\left(x\right)\cos\left(x\right)\cos\left(x\right)+\cos\left(2x\right)\sin\left(x\right)-\sin \left(x\right)}=\frac{\csc\left(x\right)}{2})
Replace with the identity =2\sin\left(x\right)\cos\left(x\right))
}{2\sin\left(x\right)\cos^2\left(x\right)+\cos\left(2x\right)\sin\left(x\right)-\sin \left(x\right)}=\frac{\csc\left(x\right)}{2})
Multiply
}{2\sin\left(x\right)\left(1-\sin^2\left(x\right)\right)+\cos\left(2x\right)\sin\left(x\right)-\sin \left(x\right)}=\frac{\csc\left(x\right)}{2})
Replace )
with )
}{2\sin\left(x\right)-2\sin^3\left(x\right)+\cos\left(2x\right)\sin\left(x\right)-\sin \left(x\right)}=\frac{\csc\left(x\right)}{2})
Distribute
}{2\sin\left(x\right)-2\sin^3\left(x\right)+\left(1-2\sin^2\left(x\right)\right)\sin\left(x\right)-\sin \left(x\right)}=\frac{\csc\left(x\right)}{2})
Replace )
with )
}{2\sin\left(x\right)-2\sin^3\left(x\right)+\sin\left(x\right)-2\sin^3\left(x\right)-\sin \left(x\right)}=\frac{\csc\left(x\right)}{2})
Distribute
}{2\sin\left(x\right)-4\sin^3\left(x\right)}=\frac{\csc\left(x\right)}{2})
Combine like terms
}{2\sin\left(x\right)\left(1-2\sin^2\left(x\right)\right)}=\frac{\csc\left(x\right)}{2})
Factor out the GCF )
\left(1\right)}=\frac{\csc\left(x\right)}{2})
Cancel like terms
}=\frac{\csc\left(x\right)}{2})
Simplify
}{2}=\frac{\csc\left(x\right)}{2})
replace with
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