Question

Open the image attached belowProve that:sec n/(tan n + cot n) = sin n

Answer 1

Given:

We are required to prove:

`\frac{\sec\text{ }\theta\text{ }}{\tan\text{ }\theta\text{ + cot}\theta}\text{ = sin}\theta`

From the left-hand side:

`\begin{gathered} =\frac{\sec\text{ }\theta\text{ }}{\tan\text{ }\theta\text{ + cot}\theta}\text{ } \\ =\text{ }\frac{(1)/(\cos\theta)}{(\sin\theta)/(\cos\theta)\text{ + }(\cos \theta)/(\sin \theta)} \\ =\text{ }((1)/(\cos\theta))/((\sin ^2\theta+cos^2\theta)/(\sin \theta\cos \theta)) \\ \end{gathered}`

From standard trigonometric identity, we have:

`\sin ^2\theta+cos^2\theta\text{ = 1}`

Substituting we have:

`\begin{gathered} =\text{ }((1)/(\cos\theta))/((1)/(\sin \theta\cos \theta)) \\ =\text{ }(\sin \theta\cos \theta)/(\cos \theta) \\ =\text{ sin }\theta\text{ (Right-hand side)} \end{gathered}`