Question

Write the trigonometric expression in terms of sine and cosine, and then simplify.

Answer 1

Answer:

$\displaystyle \frac{\tan\theta}{\sec\theta - \cos\theta} = \frac{1}{\sin\theta} = \csc\theta$

Step-by-step explanation:

We have the expression:

$\displaystyle \frac{\tan\theta}{\sec\theta - \cos\theta}$

And we want to write the expression in terms of sine and cosine and simplify.

Thus, let tanθ = sinθ / cosθ and secθ = 1 / cosθ. Substitute:

$=\displaystyle \frac{\dfrac{\sin\theta}{\cos\theta}}{\dfrac{1}{\cos\theta}-\cos\theta}$

Multiply both layers by cosθ:

$=\displaystyle \frac{\left(\dfrac{\sin\theta}{\cos\theta}\right)\cdot \cos\theta}{\left(\dfrac{1}{\cos\theta}-\cos\theta\right)\cdot \cos\theta}$

Distribute:

$\displaystyle =\frac{\sin\theta}{1-\cos^2\theta}$

Recall from the Pythagorean Theorem that sin²θ + cos²θ = 1. Hence, 1 - cos²θ = sin²θ. Substitute and simplify:

$\displaystyle =\frac{\sin\theta}{\sin^2\theta} \\ \\ =\frac{1}{\sin\theta}$

We can convert this to cosecant if we wish.