Question

Prove that sec θ cot θ = csc θ for all θ, 0o ≤ θ ≤ 360 degrees, except for 0 degrees, 90 degrees, 180 degrees, 270 degrees, and 360 degrees.

Answer 1

Explanations:

From the given information, you are to show that the product of secθ and cot θ is equivalent to cosecθ that is:

`\sec \theta cot\theta=co\sec \theta`

According to trigonometry identity:

`\begin{gathered} sec\theta=(1)/(\cos\theta) \\ cot\theta=\frac{\cos \theta}{\text{sin}\theta} \\ co\sec \theta=(1)/(\sin \theta) \end{gathered}`

Starting from the LHS, we will substitute the given identity into the expression given.

`\begin{gathered} \sec \theta cot\theta=\frac{1}{\cancel{\cos\theta}}\frac{\cancel{\cos \theta}^1}{\sin \theta} \\ \sec \theta cot\theta=1\cdot(1)/(\sin \theta) \\ \sec \theta cot\theta=(1)/(\sin \theta) \\ \sec \theta cot\theta=\text{cosec}\theta\text{ (Proved)} \end{gathered}`