2.
Determine the exact value of \(\cos\left(\frac{\pi}{3}+\frac{\pi}{4}\right)\).
Solution: Use \(\cos(a+b)=\cos a\cos b-\sin a\sin b\). Substitute exact values: \(\frac12\cdot\frac1{\sqrt2}-\frac{\sqrt3}{2}\cdot\frac1{\sqrt2}=\frac{1-\sqrt3}{2\sqrt2}\). Answer: A
3.
Identify which step is incorrect. Proof: \(\sin(\alpha+\beta)\sin(\alpha-\beta)=\sin^2\alpha-\sin^2\beta\).
Solution: Step 4 has the wrong sign. It should be \(\sin^2lpha(1-\sin^2eta)-(1-\sin^2lpha)\sin^2eta\). Answer: D
4.
Which of the following is equivalent to \(\cos^2(4x)-\sin^2(4x)\)?
Solution: Apply \(\cos^2\theta-\sin^2\theta=\cos(2\theta)\). Let \(\theta=4x\), giving \(\cos(8x)\). Answer: B
5.
Identify which step is incorrect. Proof: \(3\cos^2\theta+2\cos(2\theta)=5-7\sin^2\theta\).
Solution: Every algebraic step is valid. The proof is correct. Answer: E
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