Question

show that \\( \sec ( - x ) = \sec x \\) by writing \\( \sec ( - x ) \\) as \\( \frac { 1 } { \cos ( - x ) } \\) and then using the relationship between \\( \cos ( - x ) \\) and \\( \cos x \\).
complete the proof below.
\\( \sec ( - x ) = \frac { 1 } { \cos ( - x ) } \\) given: definition of secant
=
=
=

• \\( \frac { 1 } { \cos x } \\)

\\( \frac { 1 } { \cos x } \\)

• \\( \cos x \\)

Explanation:

Step1: Use the property of cosine function

Since \(\cos(-x)=\cos x\) (cosine is an even function), we substitute \(\cos(-x)\) with \(\cos x\) in the expression \(\frac{1}{\cos(-x)}\).
So, \(\frac{1}{\cos(-x)}=\frac{1}{\cos x}\)

Step2: Use the definition of secant function

By the definition of secant, \(\sec x=\frac{1}{\cos x}\).

Answer:

The second line of the statement is \(\frac{1}{\cos x}\) with the reason “\(\cos(-x)=\cos x\) (cosine is an even - function)” and the third line of the statement is \(\sec x\) with the reason “Definition of secant (\(\sec x = \frac{1}{\cos x}\))”