Principle Of Mathematical Induction

Question
CBSEENMA11014307

Prove that:

cos 2 straight x space equals space cos squared straight x minus sin squared straight x space equals space 1 minus 2 sin squared straight x space equals space 2 cos squared straight x minus 1 space equals space fraction numerator 1 minus tan squared straight x over denominator 1 plus tan squared straight x end fraction.

Solution

We know that
              space space space space cos left parenthesis straight alpha plus straight beta right parenthesis space equals space cosα space cosβ space minus space sinα space sinβ
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#6 {main}</pre> by x, we get
                         cos(x + x) = cosx. cosx - sinx. sinx
∴                           cos 2x = cos squared straight x minus sin squared straight x                                                           ...(A)
Also, we know that  cos squared straight x plus sin squared straight x equals 1
∴                             cos squared straight x space equals space 1 minus sin squared straight x                                                               ...(i)
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#6 {main}</pre>                                                              ...(ii)
Putting (i) in (A), we get
                                 cos space 2 straight x space equals space 1 minus sin squared straight x minus sin squared straight x
                                 cos space 2 straight x space equals space 1 minus 2 sin squared straight x                                                           ...(B)
Putting (ii) in (A), we get
                                 cos space 2 straight x space equals space cos squared straight x minus left parenthesis 1 minus cos squared straight x right parenthesis
                                  cos space 2 straight x space equals space 2 cos squared straight x minus 1                                                         ...(C)
Now,             cos space 2 straight x space equals space fraction numerator cos squared straight x minus sin squared straight x over denominator 1 end fraction equals fraction numerator cos squared straight x minus sin squared straight x over denominator cos squared straight x plus sin squared straight x end fraction equals fraction numerator begin display style fraction numerator cos squared straight x over denominator cos squared straight x end fraction end style minus begin display style fraction numerator sin squared straight x over denominator cos squared straight x end fraction end style over denominator begin display style fraction numerator cos squared straight x over denominator cos squared straight x end fraction end style plus begin display style fraction numerator sin squared straight x over denominator cos squared straight x end fraction end style end fraction          
∴              space space space space space cos space 2 straight x space equals space fraction numerator 1 minus tan squared straight x over denominator 1 plus tan squared straight x end fraction.                                                                    ....(D)
From (A), (B), (C) and (D), we have
             
                       cos space 2 straight x space equals space cos squared straight x minus sin squared straight x equals 1 minus 2 sin squared straight x equals 2 cos squared straight x minus 1 equals fraction numerator 1 minus tan squared straight x over denominator 1 plus tan squared straight x end fraction.
          

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