त्रिकोणमिति का परिचय

यदि 3 cot A = 4, तो जाँच कीजिए कि है या नहीं। - Wired Faculty

Question

यदि 3 cot A = 4, तो जाँच कीजिए कि
$fraction numerator 1 minus tan squared straight A over denominator 1 plus tan squared straight A end fraction equals cos squared straight A minus sin squared straight A$ है या नहीं।

Solution

माना कि

increment ABC

एक समकोण त्रिभुज है जिसमें

angle straight B space equals space 90 degree

.
3 cot A = 4

rightwards double arrow

cotA space equals space 4 over 3 space equals space AB over BC

माना लीजिए, AB = 4K और BC = 3K

पाइथागोरस प्रमेय के प्रयोग से:
AC² = AB² + BC²= (4K)² + (3K)²= 16K² + 9K² = 25K²अत: AC = 5K
$tan space straight A space equals space BC over AB space equals space fraction numerator 3 straight K over denominator 4 straight K end fraction space equals space 3 over 4$
अब, $cosA space equals space AB over AC equals fraction numerator 4 straight K over denominator 5 straight K end fraction space equals space 4 over 5$
और $sinA space equals space BC over AC equals fraction numerator 3 straight K over denominator 5 straight K end fraction space equals space 3 over 5$
L.H.S. = $fraction numerator 1 minus tan squared straight A over denominator 1 plus tan squared straight A end fraction$
$equals space fraction numerator 1 minus open parentheses begin display style 3 over 4 end style close parentheses squared over denominator 1 plus open parentheses begin display style 3 over 4 end style close parentheses squared end fraction space equals space fraction numerator 1 minus begin display style 9 over 16 end style over denominator 1 plus begin display style 9 over 16 end style end fraction$
$equals space fraction numerator begin display style 7 over 16 end style over denominator begin display style 25 over 16 end style end fraction equals 7 over 25$
अब,
R.H.S. = $cos squared straight A space minus space sin squared straight A$
$equals space open parentheses 4 over 5 close parentheses squared minus open parentheses 3 over 5 close parentheses squared equals space 16 over 25 minus 9 over 25 space equals space fraction numerator 16 minus 9 over denominator 25 end fraction space equals space 7 over 25$
अत: L.H.S. = R.H.S.
अथवा
$fraction numerator 1 minus tan squared straight A over denominator 1 plus tan squared straight A end fraction equals cos squared straight A space minus space sin squared straight A$