Question

I need help in math can you please help me

Answer 1

To verify the identity we need to remember the definitions of the trigonometric functions:

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

With this in mind:

`\begin{gathered} \text{tan}^2\theta-\cot ^2\theta=(\sin^2\theta)/(\cos^2\theta)-(\cos ^2\theta)/(\sin ^2\theta) \\ =(\sin ^4\theta-\cos ^4\theta)/(\cos ^2\theta\sin ^2\theta) \\ =((\sin ^2\theta+\cos ^2\theta)(\sin ^2\theta-\cos ^2\theta))/(\cos ^2\theta\sin ^2\theta) \end{gathered}`

Now we need to remembert the identity:

`\sin ^2\theta+\cos ^2\theta=1`

then:

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