Question

I’m so lost on this question i don’t even know where to start

Answer 1

The given expression is:

`\csc b(\csc b+\cot b)`

It is required to simplify the expression to a single trigonometry function.

To do this, trigonometric identities have to be applied to simplify the expression.

The following trigonometry identities will be used:

`\begin{gathered} \csc b=(1)/(\sin b) \\ \cot b=(\cos b)/(\sin b) \\ \sin^2b=1-\cos^2b \end{gathered}`

Use the inverse trigonometry identities to rewrite the expression as:

`\begin{gathered} (1)/(\sin b)((1)/(\sin b)+(\cos b)/(\sin b)) \\ Simplify\text{ the sum in parentheses:} \\ =(1)/(\sin b)((1+\cos b)/(\sin b)) \end{gathered}`

Multiply the expressions:

`(1+\cos b)/(\sin^2b)`

Rewrite the denominator using the trigonometric identity:

`(1+\cos b)/(1-\cos^2b)`

Rewrite the denominator further using the difference of two squares of binomials:

`(1+\cos b)/((1+\cos b)(1-\cos b))`

Cancel out like terms in the numerator and denominator, so the expression becomes:

`(1)/(1-\cos b)`

Hence, the required answer.