Trig expressions using fundemental trig identities

Question

I need helping figuring this out. I don't understand. Does $\sec x - \cos x = 1$? What do I do after that.

$$\sec x (\sec x -\cos x )$$

Answer 1

we have $$\sec(x)-\cos(x)=\frac{1}{\cos(x)}-\cos(x)=\frac{1-\cos(x)^2}{\cos(x)}$$ and this is not one.

Answer 2

$\sec x - \cos x$ does not equal $1$. But $\cos x = \frac{1}{\sec x}.$ So your expression equals $\sec^2 x -1$, which should be one side of a familiar trig identity.

Answer 3

Well, sec(x) multiplied by cos(x) is 1. As a beginner you should try to convert every such expression into sin and cosine to see it clearly. As your experience increases, you can do it all mentally.

Answer 4

We have: $$\sec{x}(\sec{x}-\cos{x})$$ To answer your first question, $\sec{x}-\cos{x}\neq 1$.

However, $\cos{x}=\frac{1}{\sec{x}}$.

Therefore, you can simplify this to:

$$\sec{x}(\sec{x}-\frac{1}{\sec{x}})=\sec^2{x}-1$$

Note the following identity: $\sec^2{x}-1 \equiv \tan^2{x}$

Therefore:

$$\boxed{\sec{x}(\sec{x}-\cos{x})=\tan^2{x}}$$