Given:
tanθ + cotθ = secθ * cscθ
Let's start on the right-hand side, rewriting it by means of these trigonometric identities:
[tex]\begin{gathered} \tan \theta=\frac{\sin\theta}{\cos\theta} \\ \cot \theta=\frac{\cos\theta}{\sin\theta} \end{gathered}[/tex]Then, we get:
[tex]\begin{gathered} \tan \theta+\cot \theta=\sec \theta\times\csc \theta \\ \frac{\sin\theta}{\cos\theta}+\frac{\cos\theta}{\sin\theta}=\sec \theta\times\csc \theta \end{gathered}[/tex]Now, let's divide and multiply the first term on the right-hand side by sinθ, like this:
[tex]\begin{gathered} \frac{\sin\theta}{\cos\theta}\times\frac{\sin\theta}{\sin\theta}+\frac{\cos\theta}{\sin\theta}=\sec \theta\times\csc \theta \\ \frac{\sin^2\theta}{\cos\theta\times\sin\theta}+\frac{\cos\theta}{\sin\theta}=\sec \theta\times\csc \theta \end{gathered}[/tex]And let's do the same with the second term but with cosθ, like this:
[tex]\begin{gathered} \frac{\sin^2\theta}{\cos\theta\times\sin\theta}+\frac{\cos\theta}{\sin\theta}\times\frac{\cos\theta}{\cos\theta}=\sec \theta\times\csc \theta \\ \frac{\sin^2\theta}{\cos\theta\times\sin\theta}+\frac{\cos^2\theta}{\cos\theta\times\sin\theta}=\sec \theta\times\csc \theta \end{gathered}[/tex]Now that both terms on the right-hand side have the same denominator, let's sum the numerators, like this:
[tex]\frac{\sin^2\theta+\cos^2\theta}{\cos\theta\times\sin\theta}=\sec \theta\times\csc \theta[/tex]By means of the trigonometric identity:
[tex]\sin ^2\theta+\cos ^2\theta=1[/tex]We can rewrite the above expression like this
[tex]\frac{1}{\cos\theta\times\sin\theta}=\sec \theta\times\csc \theta[/tex]And now, we can separate the denominators, expressing the right side of the equation as a product of fractions, like this:
[tex]\begin{gathered} \frac{1}{\cos\theta\times\sin\theta}=\sec \theta\times\csc \theta \\ \frac{1}{\cos\theta}\times\frac{1}{\sin\theta}=\sec \theta\times\csc \theta \end{gathered}[/tex]From the trigonometric identities:
[tex]\begin{gathered} \sec \theta=\frac{1}{\cos \theta} \\ \csc \theta=\frac{1}{\sin \theta} \end{gathered}[/tex]Then, we can rewrite the above expression, like this:
[tex]\sec \theta\times\csc \theta=\sec \theta\times\csc \theta[/tex]Proved!
