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Some Applications Of Trigonometry

Question
CBSEENMA10007613
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If 3 cot A = 4, check whether
fraction numerator 1 minus tan squared straight A over denominator 1 plus tan squared straight A end fraction equals cos squared straight A space minus space sin squared straight A or not.

Solution
Let us draw a right triangle right angled at B.
We have,
3 cot A = 4
rightwards double arrow                 cotA space equals space 4 over 3 space equals space AB over BC
Let AB = 4K and BC = 3K
WiredFaculty

Using Pythagoras theorem, we have
AC2 = AB2 + BC2
= (4K)2 + (3K)2
= 16K2 + 9K2 = 25 K2
So, AC = 5K
tan space straight A space equals space BC over AB space equals space fraction numerator 3 straight K over denominator 4 straight K end fraction space equals space 3 over 4
Now, WiredFaculty
and sinA space equals space BC over AC equals fraction numerator 3 straight K over denominator 5 straight K end fraction space equals space 3 over 5
     L.H.S. = WiredFaculty
              space space equals space fraction numerator 1 minus open parentheses begin display style 3 over 4 end style close parentheses squared over denominator 1 plus open parentheses begin display style 3 over 4 end style close parentheses squared end fraction space equals space fraction numerator 1 minus begin display style 9 over 16 end style over denominator 1 plus begin display style 9 over 16 end style end fraction
                 WiredFaculty
Now, 
R.H.S. = cos squared straight A space minus space sin squared straight A
           WiredFaculty
Hence, L.H.S. = R.H.S.
or space fraction numerator 1 minus tan squared straight A over denominator 1 plus tan squared straight A end fraction equals cos squared straight A space minus space sin squared straight A.

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