Question

Write each expression in terms of sines and or cosines and then simplify sec x and cos x

Answer 1

To write each expression in terms of sines and/or cosines:

1. sec x:
Recall that sec x = 1/cos x. Therefore, sec x can be written as:

sec x = 1/cos x

2. cos x:
Since cos x is already in terms of cosines, there is no need to rewrite it.

Now, simplifying:

1. sec x:
To simplify sec x, we can rationalize the denominator by multiplying both the numerator and denominator by the conjugate of cos x:

sec x = (1/cos x) * (cos x/cos x)
= cos x/cos^2 x
= cos x/(1 - sin^2 x) [using the Pythagorean identity: cos^2 x = 1 - sin^2 x]

Therefore, sec x simplified is cos x/(1 - sin^2 x).

2. cos x:
There is no need to simplify cos x as it is already a simplified expression.

So, the simplified expressions are:
sec x = cos x/(1 - sin^2 x)
cos x = cos x