tan(θ)/ sec(θ) − cos(θ)

Please help!

tan(θ)/ sec(θ) − cos(θ)

= tanθ * cosθ - cosθ
= sinθ/cosθ * cosθ - cosθ
= sinθ - cosθ

To simplify the expression tan(θ)/sec(θ) - cos(θ), we will use trigonometric identities.

Let's start by rewriting sec(θ) as 1/cos(θ). So, our expression becomes (tan(θ))/(1/cos(θ)) - cos(θ).

The next step is to simplify the expression by combining the fractions. To do this, we multiply the numerator and denominator of the first fraction, tan(θ), by cos(θ). This will give us (tan(θ) * cos(θ))/ (1 - cos(θ) * cos(θ)) - cos(θ).

The numerator, tan(θ) * cos(θ), can be simplified using the identity tan(θ) = sin(θ)/cos(θ).

So, the new numerator becomes (sin(θ) * cos(θ))/ (1 - cos(θ) * cos(θ)) - cos(θ).

Now, let's expand the denominator, (1 - cos(θ) * cos(θ)). Using the identity sin^2(θ) + cos^2(θ) = 1, we can rewrite the denominator as sin^2(θ).

Therefore, the expression can be further simplified as follows:
(sin(θ) * cos(θ))/ sin^2(θ) - cos(θ).

Next, cancel out the common factor, cos(θ), in the numerator and denominator. This leaves us with sin(θ)/sin^2(θ).

Lastly, simplify sin(θ)/sin^2(θ). Recall that sin(θ)/sin^2(θ) is equal to 1/sin(θ).

So, tan(θ)/sec(θ) - cos(θ) simplifies to 1/sin(θ) - cos(θ).