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:

$\begin{gathered} \csc b=\frac{1}{\sin b} \\ \cot b=\frac{\cos b}{\sin b} \\ \sin^2b=1-\cos^2b \end{gathered}$

Use the inverse trigonometry identities to rewrite the expression as:

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

Multiply the expressions:

$\frac{1+\cos b}{\sin^2b}$

Rewrite the denominator using the trigonometric identity:

$\frac{1+\cos b}{1-\cos^2b}$

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

$\frac{1+\cos b}{(1+\cos b)(1-\cos b)}$

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

$\frac{1}{1-\cos b}$

Hence, the required answer.