Answer:
Proved See below
Step-by-step explanation:
Man this one is a world of its own :D Just a quick question are you a fellow Add Math student in O levels i remember this question from back in the day :D Anyhow Lets get started
For this question we need to know the following identities:
[tex]1+tan^{2}x=sec^2x\\\\1+cot^2x=cosec^2x\\\\sin^2x+cos^2x=1[/tex]
Lets solve the bottom most part first:
[tex]1-\frac{1}{1-sec^2x} \\\\[/tex]
Take LCM
[tex]1-\frac{1}{1-sec^2x} \\\\\frac{1-sec^2x-1}{1-sec^2x} \\\\\frac{-sec^2x}{1-sec^2x} \\\\\frac{-(1+tan^2x)}{-tan^2x}[/tex]
now break the LCM
[tex]\frac{-1}{-tan^2x}+\frac{-tan^2x}{-tan^2x}\\\\\frac{1}{tan^2x}+1\\\\cot^2x+1[/tex]
because 1/tan = cot x
and furthermore,
[tex]cot^2x+1\\cosec^2x[/tex]
now we solve the above part and replace the bottom most part that we solved with [tex]cosec^2x[/tex]
[tex]\frac{1}{1-\frac{1}{cosec^2x} } \\\\\frac{1}{1-sin^2x} \\\\\frac{1}{cos^2x}\\\\sec^2x[/tex]
Hence proved! :D
