Question

I need help in math can you please help me

Answer 1

We have to work with left-hand side and make it same as right-hand side.

Let's do it:

`(\sin \theta)/(1-\cos \theta)`
`(\sin\theta)/(1-\cos\theta)*(1+\cos\theta)/(1+\cos\theta)=\frac{\sin \theta(1+\cos \theta)}{(1-\cos \theta)(1+\cos \theta)_{}}`

Now, simplifying further:

`\begin{gathered} \frac{\sin\theta(1+\cos\theta)}{(1-\cos\theta)(1+\cos\theta)_{}} \\ =(\sin\theta+\sin\theta\cos\theta)/(1-\cos^2\theta) \end{gathered}`

We know the identity:

`\begin{gathered} \sin ^2\theta+\cos ^2\theta=1 \\ or \\ \sin ^2\theta=1-\cos ^2\theta \end{gathered}`

Substituting, we get:

`\begin{gathered} (\sin\theta+\sin\theta\cos\theta)/(1-\cos^2\theta) \\ =(\sin\theta+\sin\theta\cos\theta)/(\sin^2\theta) \\ =(\sin\theta)/(\sin^2\theta)+(\sin\theta\cos\theta)/(\sin^2\theta) \\ =(1)/(\sin\theta)+(\cos\theta)/(\sin\theta) \end{gathered}`

We know:

`(1)/(\sin\theta)=\csc \theta`

and

`\begin{gathered} (\sin\theta)/(\cos\theta)=\tan \theta \\ \text{and} \\ (\cos\theta)/(\sin\theta)=(1)/(\tan\theta)=\cot \theta \end{gathered}`

Of couse, we can see that it is proved.

`\begin{gathered} (1)/(\sin\theta)+(\cos\theta)/(\sin\theta) \\ =\csc \theta+\cot \theta \end{gathered}`