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Chapter 1: Problem 1

Prove the following identities using only the definitions of the trigonometric functions, the Pythagorean identity, or the identities for sines and cosines of sums of angles. a. \(\cos 2 x=2 \cos ^{2} x-1\) b. \(\sin 3 x=A \sin ^{3} x+B \sin x\), for what values of \(A\) and \(B\) ? c. \(\sec \theta+\tan \theta=\tan \left(\frac{\theta}{2}+\frac{\pi}{4}\right)\)

Short Answer

Expert verified
a. True by the cosine double angle formula. b. \(A = -4\) and \(B = 3\) by the triple angle formula. c. True by the half-angle and tangent addition formulas.

Step by step solution

01

Part a: Applying Cosine Double Angle Formula

Use the identity \(\cos 2x = 2\cos^{2}x - 1\), which is a direct application of the cosine double angle formula.
02

Part b: Applying the Triple Angle Formula

\(\sin 3x = \sin (2x+x)\). Apply the sine addition formula \(\sin(a + b) = \sin a \cos b + \cos a \sin b\). In this case, \(a = 2x\) and \(b = x\), so \(\sin 3x = \sin 2x \cos x + \cos 2x \sin x\). Apply the double angle formulas for sine and cosine, resulting in \(3 \sin x - 4 \sin^{3} x\). So, \(A = -4\) and \(B = 3\).
03

Part c: Applying Half-Angle and Addition Formulas

Firstly, rewrite the right hand side using the tangent addition formula, \(\tan(a + b) = (\tan a + \tan b) / (1 - \tan a \tan b)\). Then, replace \(\tan \theta/2\) with its equivalence in terms of secant from the half-angle formula. Simplify to obtain \(sec \theta + tan \theta\).

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