Principle of Mathematical Induction

Question

Prove that:

$tan open parentheses straight x over 2 close parentheses equals plus-or-minus square root of fraction numerator 1 minus cosx over denominator 1 plus cosx end fraction end root.$

Solution

We know that:
$cos space 2 straight A space equals space fraction numerator 1 minus tan squared straight A over denominator 1 plus tan squared straight A end fraction$
Replacing A by $space space straight x over 2 comma$ we get
 $cosx space equals space fraction numerator 1 minus tan squared open parentheses begin display style straight x over 2 end style close parentheses over denominator 1 plus tan squared open parentheses begin display style straight x over 2 end style close parentheses end fraction$
$rightwards double arrow$ $fraction numerator 1 plus tan squared open parentheses begin display style straight x over 2 end style close parentheses over denominator 1 minus tan squared open parentheses begin display style straight x over 2 end style close parentheses end fraction equals 1 over cosx$
Apply componendo and dividendo, we get
$rightwards double arrow$ $fraction numerator 1 plus tan squared open parentheses begin display style straight x over 2 end style close parentheses plus 1 minus tan squared open parentheses begin display style straight x over 2 end style close parentheses over denominator 1 plus tan squared open parentheses begin display style straight x over 2 end style close parentheses minus 1 plus tan squared open parentheses begin display style straight x over 2 end style close parentheses end fraction equals fraction numerator 1 plus cosx over denominator 1 minus cosx end fraction$ $rightwards double arrow$ $fraction numerator 2 over denominator 2 space tan squared open parentheses begin display style straight x over 2 end style close parentheses end fraction equals fraction numerator 1 plus cosx over denominator 1 minus cosx end fraction$
$rightwards double arrow$ $space space fraction numerator 1 over denominator tan squared open parentheses begin display style straight x over 2 end style close parentheses end fraction equals fraction numerator 1 plus cosx over denominator 1 minus cosx end fraction$ $rightwards double arrow$ $tan squared open parentheses straight x over 2 close parentheses space equals space fraction numerator 1 minus cosx over denominator 1 plus cosx end fraction$
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Principle Of Mathematical Induction

Prove that:

$tan open parentheses straight x over 2 close parentheses equals plus-or-minus square root of fraction numerator 1 minus cosx over denominator 1 plus cosx end fraction end root.$

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