1.
Determine the exact value of \(\cos\left(\frac{\pi}{3}+\frac{\pi}{4}\right)\).
Solution: Use \(\cos(x+y)=\cos x\cos y-\sin x\sin y\). Substitute exact values to obtain \(\frac{1}{2\sqrt2}-\frac{\sqrt3}{2\sqrt2}=\frac{1-\sqrt3}{2\sqrt2}\). Answer: D
2.
Identify which step is incorrect in the proof \(3\cos^2\theta+2\cos(2\theta)=5-7\sin^2\theta\).
Solution: Step 7 changes the sign incorrectly; it should be \(-1+\cos^2eta+\cos^2lpha\). Answer: G
3.
Which of the following is equivalent to \(\cos^2(4x)-\sin^2(4x)\)?
Solution: Apply the double-angle identity \(\cos^2\theta-\sin^2\theta=\cos2\theta\) with \(\theta=4x\). Therefore the result is \(\cos(8x)\). Answer: A
4.
Identify the first incorrect step in the proof \(\cos(\alpha+\beta)\cos(\alpha-\beta)=\cos^2\alpha-\sin^2\beta\).
Solution: Every step is valid; the proof is correct. Answer: E
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