Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 1 - Functions - 1.4 Trigonometric Functions and Their Inverse - 1.4 Exercises: 34

Answer

\[ = - \sec x\]

Work Step by Step

\[\begin{gathered} Using\,\,the\,trigonometric\,identity\,for\,the\,cosine\,of\,a\,sum \hfill \\ \cos \,\,\left( {a + b} \right) = \cos a\cos b - \sin a\sin b \hfill \\ \hfill \\ therefore \hfill \\ \hfill \\ \sec \,\left( {x + \pi } \right) = \frac{1}{{\cos \,\left( {x + \pi } \right)}} \hfill \\ \hfill \\ {\text{Simplify}} \hfill \\ \hfill \\ \sec \,\left( {x + \pi } \right) = \frac{1}{{\cos \,\left( x \right)\cos \,\left( \pi \right) - \sin \,\left( x \right)\sin \,\left( \pi \right)}} \hfill \\ \hfill \\ \sec \,\left( {x + \pi } \right) = \frac{1}{{\cos \,\left( x \right) \cdot \,\left( { - 1} \right) - \sin \,\left( x \right) \cdot 0}} = \frac{1}{{ - \cos \,\left( x \right)}} = - \sec x \hfill \\ \hfill \\ which\,was\,needed\,to\,be\,shown. \hfill \\ \end{gathered} \]
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