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Answer: Rational functions of trigonometric functions An
Chapter 4, Problem 72E(choose chapter or problem)
Rational functions of trigonometric functions An integrand with trigonometric functions in the numerator and denominator can often be converted to a rational integrand using the substitution u=tan(x/2) or \(x=2 \tan ^{-1} u\). The following relations are used in making this change of variables.
A: \(d x=\frac{2}{1+u^{2}} d u\)
B: \(\sin x=\frac{2 u}{1+u^{2}}\)
C: \(\cos x=\frac{1-u^{2}}{1+u^{2}}\)
Verify relation A by differentiating \(x=2 \tan ^{-1} u\). Verify relations B and C using a right-triangle diagram and the double-angle formulas
\(\sin x=2 \sin \left(\frac{x}{2}\right) \cos \left(\frac{x}{2}\right)\) and \(\cos x=2 \cos ^{2}\left(\frac{x}{2}\right)-1\).
Questions & Answers
QUESTION:
Rational functions of trigonometric functions An integrand with trigonometric functions in the numerator and denominator can often be converted to a rational integrand using the substitution u=tan(x/2) or \(x=2 \tan ^{-1} u\). The following relations are used in making this change of variables.
A: \(d x=\frac{2}{1+u^{2}} d u\)
B: \(\sin x=\frac{2 u}{1+u^{2}}\)
C: \(\cos x=\frac{1-u^{2}}{1+u^{2}}\)
Verify relation A by differentiating \(x=2 \tan ^{-1} u\). Verify relations B and C using a right-triangle diagram and the double-angle formulas
\(\sin x=2 \sin \left(\frac{x}{2}\right) \cos \left(\frac{x}{2}\right)\) and \(\cos x=2 \cos ^{2}\left(\frac{x}{2}\right)-1\).
ANSWER:SOLUTIONStep 1We are given that Therefore on differentiation we get Thus we have verified statement A.