O'Quinn Ross
10/31/2023 · Senior High School
1. Using double angle formulas, show that a) \( \frac{1}{2} \csc (A) \sec (A)=\csc (2 A) \) 2. Using half - angle formulas, show that a) \( \left.-2 \sin \left(\frac{x}{2}\right) \cos \left(\frac{x}{2}\right)=\tan (x)\right] \) b) \( \sin ^{2}\left(\frac{x}{2}\right)+\cos ^{2}\left(\frac{x}{2}\right)=1 \) Write each product as a sum or difference of sines and/or cosines. a) \( 5 \sin (4 x) \sin (6 x) \) b) \( \cos \left(\frac{2 x}{3}\right) \cos \left(\frac{4 x}{3}\right) \) Write each expression as a product of sines and/or cosines. a) \( \cos (5 x)+\cos (3 x) \) b) \( \sin \left(\frac{x}{2}\right)-\sin \left(\frac{5 x}{2}\right) \) 3.
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1. a) \( \frac{1}{2} \csc(A) \sec(A) = \csc(2A) \) is proven using the double angle formula for sine.
2. a) The statement \( -2 \sin\left(\frac{x}{2}\right) \cos\left(\frac{x}{2}\right) = \tan(x) \) seems to be incorrect. The correct expression is \( -2 \sin\left(\frac{x}{2}\right) \cos\left(\frac{x}{2}\right) = -\tan(x) \cos(x) \).
b) \( \sin^2\left(\frac{x}{2}\right) + \cos^2\left(\frac{x}{2}\right) = 1 \) is a direct application of the Pythagorean identity.
3. a) \( 5 \sin(4x) \sin(6x) = \frac{5}{2} [\cos(2x) - \cos(10x)] \)
b) \( \cos\left(\frac{2x}{3}\right) \cos\left(\frac{4x}{3}\right) = \frac{1}{2} [\cos(2x) + \cos\left(\frac{2x}{3}\right)] \)
4. a) \( \cos(5x) + \cos(3x) = 2 \cos(4x) \cos(x) \)
b) \( \sin\left(\frac{x}{2}\right) - \sin\left(\frac{5x}{2}\right) = -2 \cos(3x) \sin(2x) \)
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