Question

How do you simplify `(sin theta csc theta)/cot theta`?

Answer 1

`color(green)(tan theta`

Explanation:

From the above trigonometric identities,

`csc theta = (1 / sin theta), cot theta = 1/ tan theta = cos theta / sin theta`

Substituting the above in the given sum,

`(cancel sin theta * (1/ cancel sin theta) ) / (cos theta / sin theta)`

`=> sin theta / cos theta = color(green)(tan theta`

Answer 2

`(sin theta csc theta)/cot theta = tan theta`.

Explanation:

Firstly, the cosecant function is defined as the inverse sine function:
`csc varphi = 1/sin varphi`
and the cotangent function as the inverse tangent function: `cot varphi = 1/tanvarphi = 1/(sinvarphi/cosvarphi) = cos varphi/sin varphi`.

Substituting it into the equation, we have

`(sin theta * 1/color(red)(sin theta))/color(red)(cos theta/sintheta)`

We know that the product of a number and its inverse is always `1` :

`color(red)1/color(red)(costheta/sintheta) = sin theta/cos theta = tan theta`.