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| 1. |
Cot + cosec -1 ÷ cot-cosec- cosec + 1 = 1+ cos ÷1 +sin |
| Answer» We have,L.H.S. =\xa0{tex}\\frac { ( \\cot \\theta + cosec \\theta ) - 1 } { ( \\cot \\theta - cosec \\theta + 1 ) }{/tex}{tex}= \\frac { ( cosec \\theta + \\cot \\theta ) - \\left( cosec ^ { 2 } \\theta - \\cot ^ { 2 } \\theta \\right) } { ( \\cot \\theta - cosec \\theta + 1 ) }{/tex}\xa0[{tex}\\because{/tex} 1 = cosec2{tex}\\theta{/tex} - cot2{tex}\\theta{/tex}]{tex}= \\frac { ( cosec \\theta + \\cot \\theta ) [ 1 - ( cosec \\theta - \\cot \\theta ) ] } { ( \\cot \\theta - cosec \\theta + 1 ) }{/tex}{tex}= \\frac { ( cosec \\theta + \\cot \\theta ) ( \\cot \\theta - cosec \\theta + 1 ) } { ( \\cot \\theta - cosec \\theta + 1 ) }{/tex}{tex}= ( cosec \\theta + \\cot \\theta ) = \\left( \\frac { 1 } { \\sin \\theta } + \\frac { \\cos \\theta } { \\sin \\theta } \\right) = \\frac { 1 + \\cos \\theta } { \\sin \\theta }{/tex}= R.H.S.Hence,\xa0{tex}\\frac { \\cot \\theta + \\ cosec \\theta - 1 } { \\cot \\theta - \\ cosec \\theta + 1 } = \\frac { 1 + \\cos \\theta } { \\sin \\theta }{/tex}. | |