Question

I need help in math can you please help me 0

Answer 1

Answer:

We will use the following identities:

$\begin{gathered} \csc (x)=\frac{1}{\sin (x)} \\ \cot (x)=\frac{\cos (x)}{\sin (x)} \end{gathered}$

So, replacing the identities on the left side, we get:

$\frac{1+\csc^2(x)}{\cot^2(x)+1}=\frac{1+(\frac{1}{\sin(x)})^2}{(\frac{\cos (x)}{\sin (x)})^2+1}$

Then, solving the power and adding the expressions, we get:

$\frac{1+csc^2(x)}{\cot^2(x)+1}=\frac{1+\frac{1}{\sin^2(x)}}{\frac{\cos^2(x)}{\sin^2(x)}+1}$$\frac{1+csc^2(x)}{\cot^2(x)+1}=\frac{\frac{\sin^2(x)+1}{\sin^2(x)}}{\frac{\cos ^2(x)+\sin ^2(x)}{\sin ^2(x)}}$

Dividing the expression and simplifying, we get:

$\frac{1+csc^2(x)}{\cot^2(x)+1}=\frac{\sin ^2(x)+1}{\cos ^2(x)+\sin ^2(x)}$

Finally, we know that cos²(x) + sin²(x) = 1, so we can rewrite the expression as:

$\begin{gathered} \frac{1+\csc^2(x)}{\cot^2(x)+1}=\frac{\sin ^2(x)+1}{1} \\ \frac{1+\csc^2(x)}{\cot^2(x)+1}=\sin ^2(x)+1 \end{gathered}$