Prove that:tan theta – 1 + sec theta / tan theta + 1-sec theta = 1 / sec theta – tan theta
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L.H.S={tex}\\frac { \\tan \\theta + \\sec \\theta – 1 } { \\tan \\theta – \\sec \\theta + 1 }{/tex}{tex}= \\frac { ( \\tan \\theta + \\sec \\theta ) – \\left( \\sec ^ { 2 } \\theta – \\tan ^ { 2 } \\theta \\right) } { \\tan \\theta – \\sec \\theta + 1 }{/tex}\xa0{tex}[\\because sec^2\\theta-tan^2\\theta=1]{/tex}{tex}= \\frac { ( \\tan \\theta + \\sec \\theta ) – ( \\sec \\theta – \\tan \\theta ) ( \\sec \\theta + \\tan \\theta ) } { \\tan \\theta – \\sec \\theta + 1 }{/tex}{tex}= \\frac { ( \\tan \\theta + \\sec \\theta ) [ 1 – \\sec \\theta + \\tan \\theta ] } { \\tan \\theta – \\sec \\theta + 1 }{/tex}=\xa0{tex}tan\\theta+sec\\theta{/tex}= R.H.S.Hence Proved.