# Prove that : `(1+sin a-cos a)/(1+sin a+cos a) + (1+sin a + cos a)/(1+sin a -cos a)=2*cosec a`

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The identity `(1+sin a-cos a)/(1+sin a+cos a) + (1+sin a + cos a)/(1+sin a -cos a)=2*cosec a` has to be proved

`(1+sin a-cos a)/(1+sin a+cos a) + (1+sin a + cos a)/(1+sin a -cos a)`

=> `((1+sin a-cos a)^2 + (1+sin a + cos a)^2)/((1+sin a)^2 -(cos a)^2)`

=> `(1 + sin^2 a + cos^2 a + 2*sin +1 + sin^2 a + cos^2 a + 2*sin)/((1+sin a)^2 -(cos a)^2)`

=> `(1 + 1 + 2*sina +1 + 1 + 2*sina)/((1+sin a)^2 -(cos a)^2)`

=> `(4 + 4*sina)/((1+sin a)^2 -(cos a)^2)`

=> `(4*(1 + sin a))/((1+sin a)^2 -(cos a)^2)`

=> `(4*(1 + sin a))/(1+sin^2a+2*sin a -cos^2 a)`

=> `(4*(1 + sin a))/(2*sin^2a+2*sin a)`

=> `(4*(1 + sin a))/(2*sin a(1 + sin a))`

=> `2/sin a`

=> `2*cosec a`

This proves the identity `(1+sin a-cos a)/(1+sin a+cos a) + (1+sin a + cos a)/(1+sin a -cos a)=2*cosec a`

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