Principle Of Mathematical Induction

Question
CBSEENMA11014312

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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Some More Questions From Principle of Mathematical Induction Chapter

Find 1 radian angle in degrees, minutes, seconds.