Question

Simplify sin x(cot x-csc x)+1

Answer 1

`\displaystyle{\sin x \left(\cot x - \csc x\right)+1 = \cos x}`

Explanation:

The following expression can be rewritten as:

`\displaystyle{\sin x \left((\cos x)/(\sin x) -(1)/(\sin x)\right) +1}`

We know that:

`\displaystyle{\cot x = (1)/(\tan x)}`

and also:

`\displaystyle{\tan x = (\sin x )/(\cos x)}`

Hence:

`\displaystyle{\cot x = (1)/( (\sin x)/(\cos x)) = (\cos x)/(\sin x)}`

We also know that:

`\displaystyle{\csc x = (1)/(\sin x)}`

Thus, how we can rewrite into the form of:

`\displaystyle{\sin x \left((\cos x)/(\sin x) -(1)/(\sin x)\right) +1}`

Expand sin(x) in:

`\displaystyle{\sin x \cdot (\cos x)/(\sin x) + \sin x \cdot \left(-(1)/(\sin x)\right) + 1}\\\\\displaystyle{= \cos x - 1 + 1}\\\\\displaystyle{= \cos x}`

Thus:

`\displaystyle{\sin x \left(\cot x - \csc x\right)+1 = \cos x}`