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Answer:
[tex]\displaystyle \frac{\tan\theta}{\sec\theta - \cos\theta} = \frac{1}{\sin\theta} = \csc\theta[/tex]
Step-by-step explanation:
We have the expression:
[tex]\displaystyle \frac{\tan\theta}{\sec\theta - \cos\theta}[/tex]
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:
[tex]=\displaystyle \frac{\dfrac{\sin\theta}{\cos\theta}}{\dfrac{1}{\cos\theta}-\cos\theta}[/tex]
Multiply both layers by cosθ:
[tex]=\displaystyle \frac{\left(\dfrac{\sin\theta}{\cos\theta}\right)\cdot \cos\theta}{\left(\dfrac{1}{\cos\theta}-\cos\theta\right)\cdot \cos\theta}[/tex]
Distribute:
[tex]\displaystyle =\frac{\sin\theta}{1-\cos^2\theta}[/tex]
Recall from the Pythagorean Theorem that sin²θ + cos²θ = 1. Hence, 1 - cos²θ = sin²θ. Substitute and simplify:
[tex]\displaystyle =\frac{\sin\theta}{\sin^2\theta} \\ \\ =\frac{1}{\sin\theta}[/tex]
We can convert this to cosecant if we wish.