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ગણિત ધોરણ 11 સેમિસ્ટર 2

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સાબિત કરો :
fraction numerator 1 space plus space sin space 2 straight theta space plus space cos space 2 straight theta over denominator 1 space plus space sin space 2 straight theta space minus space cos space 2 straight theta end fraction space equals space cot space straight theta


straight L. straight H. straight S. space equals space fraction numerator 1 space plus space sin space 2 straight theta space plus space cos space 2 straight theta over denominator 1 space plus space sin space 2 straight theta space minus space cos space 2 straight theta end fraction


         equals space fraction numerator 1 space plus space 2 space sin space straight theta space cos space straight theta space plus space 2 space cos squared space straight theta space minus space 1 over denominator 1 space plus space 2 space sin space straight theta space cos space straight theta space minus space open parentheses 1 space minus space 2 space sin squared space straight theta close parentheses end fraction


equals space fraction numerator 2 space sin space straight theta space cos space straight theta space plus space 2 space cos squared space straight theta over denominator 2 space sin space straight theta space cos space straight theta space plus space 2 space sin squared space straight theta end fraction


equals space fraction numerator cos space straight theta space open parentheses 2 space sin space straight theta space plus space 2 space cos space straight theta close parentheses over denominator sin space straight theta space open parentheses 2 space cos space straight theta space plus space 2 space sin space straight theta close parentheses end fraction


equals space fraction numerator cos space straight theta over denominator sin space straight theta end fraction


equals space cot space straight theta


equals space straight R. straight H. straight S.

સાબિત કરો :
tan space straight theta over 2 space plus space cot space straight theta over 2 space equals space 2 space cosec space straight theta


straight L. straight H. straight S. space equals space tan space straight theta over 2 space plus space cot space straight theta over 2


         equals space fraction numerator sin space straight theta over 2 over denominator cos space straight theta over 2 end fraction space plus space fraction numerator cos space straight theta over 2 over denominator sin space straight theta over 2 end fraction


equals space fraction numerator sin squared space begin display style straight theta over 2 end style space plus space cos squared space begin display style straight theta over 2 end style over denominator sin space begin display style straight theta over 2 end style space cos space begin display style straight theta over 2 end style end fraction


equals space fraction numerator 2 space cross times space 1 over denominator 2 space sin space straight theta over 2 space cos space straight theta over 2 end fraction


equals space fraction numerator 2 over denominator sin space straight theta end fraction

equals space 2 space cosec space straight theta


equals space straight R. straight H. straight S.

સાબિત કરો :
fraction numerator cos space 2 straight theta over denominator 1 space plus space sin space 2 straight theta end fraction space equals space cot space open parentheses straight pi over 4 space plus space straight theta close parentheses


straight L. straight H. straight S. space equals space fraction numerator cos space 2 straight theta over denominator 1 space plus space sin space 2 straight theta end fraction


         equals space fraction numerator sin space open parentheses straight pi over 2 space minus space 2 straight theta close parentheses over denominator 1 space plus space cos space open parentheses straight pi over 2 space minus space 2 straight theta close parentheses end fraction


equals space fraction numerator 2 space sin space open parentheses begin display style straight pi over 4 end style space minus space straight theta close parentheses space cos space open parentheses begin display style straight pi over 4 end style space minus space straight theta close parentheses over denominator 2 space cos squared space open parentheses begin display style straight pi over 4 end style space minus space straight theta close parentheses end fraction


equals space fraction numerator sin space open parentheses begin display style straight pi over 4 end style space minus space straight theta close parentheses over denominator cos space open parentheses begin display style straight pi over 4 end style space minus space straight theta close parentheses end fraction


equals space tan space open parentheses straight pi over 4 space minus space straight theta close parentheses


equals space cot space open parentheses straight pi over 2 space minus space open parentheses straight pi over 4 space minus space straight theta close parentheses close parentheses


equals space cot space open parentheses straight pi over 4 space plus space straight theta close parentheses

equals space straight R. straight H. straight S.

સાબિત કરો :
fraction numerator sin space 2 straight theta over denominator 1 space minus space cos space 2 straight theta end fraction space equals space cot space straight theta


straight L. straight H. straight S. space equals space fraction numerator sin space 2 straight theta over denominator 1 space minus space cos space 2 straight theta end fraction

space space space space space space space space
space space space space space space space space space space space space space equals space fraction numerator 2 space sin space straight theta space cosθ over denominator 2 space sin squared space straight theta space end fraction


space space space space space space space space space space space space space equals space fraction numerator cos space straight theta over denominator sin space straight theta end fraction space equals space cot space space straight theta space equals space straight R. straight H. straight S.

સાબિત કરો :
fraction numerator cos space straight theta over denominator 1 space plus space sin space straight theta end fraction space equals space tan space open parentheses straight pi over 2 space minus space theta over 2 close parentheses


straight L. straight H. straight S. space equals space fraction numerator cos space straight theta over denominator 1 space plus space sin space straight theta end fraction


         equals space fraction numerator begin display style fraction numerator 1 space minus space tan squared space straight theta over 2 over denominator 1 space plus space tan squared space straight theta over 2 end fraction end style over denominator space 1 space plus space begin display style fraction numerator 2 space tan space straight theta over 2 over denominator 1 space plus space tan squared space straight theta over 2 end fraction end style end fraction



equals space fraction numerator 1 space minus space tan squared space straight theta over 2 over denominator 1 space plus space tan squared space straight theta over 2 space plus space 2 space tan space straight theta over 2 end fraction


equals space fraction numerator open parentheses 1 space minus space tan space straight theta over 2 close parentheses space open parentheses 1 space plus space tan space straight theta over 2 close parentheses over denominator open parentheses 1 space plus space tan space straight theta over 2 close parentheses squared end fraction


equals space fraction numerator 1 space minus space tan space straight theta over 2 over denominator 1 space plus space tan space straight theta over 2 end fraction


equals space fraction numerator tan space straight pi over 4 space minus space tan space straight theta over 2 over denominator space 1 space plus space tan space straight pi over 4 times space tan space straight theta over 2 space end fraction


equals space tan space open parentheses straight pi over 4 space minus space straight theta over 2 close parentheses


equals space straight R. straight H. straight S. space