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Question

Prove that: $$\dfrac { \tan\theta +\sec\theta -1 }{ \tan\theta -\sec\theta -1 } =\sec\theta +\tan\theta \\ $$


Solution

LHS $$=\dfrac { \tan\theta +\sec\theta -1 }{ \tan\theta -\sec\theta -1 } $$
        $$=\dfrac { \tan\theta +\sec\theta -{ \sec }^{ 2 }\theta +{ \tan }^{ 2 }\theta  }{ \tan\theta -\sec\theta  } $$
        $$=\dfrac { \left( \tan\theta +\sec\theta  \right) -\left( \sec\theta +\tan\theta  \right) \left( \sec\theta -\tan\theta  \right)  }{ \left( \tan\theta -\sec\theta  \right)  } $$
        $$=\dfrac { \left( \tan\theta +\sec\theta  \right) \left( 1-\sec\theta +\tan\theta  \right)  }{ \left( \tan\theta -\sec\theta +1 \right)  } $$
        $$=\tan\theta +\sec\theta $$ (RHS)
                                     (proved)

Mathematics

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