Question

Prove this identity without proving RHS

(Cosec-sec)(cot-tan)= (cosec+sec)(sec.cosec-2)

Answer 1

taking L.H.S

=( cosec- sec)(cot- tan)

= ( 1/sin -1/cos) (cos/sin-sin/cos)

=cos/sin cos-sin/sin cos)(cos^2/sin cos-sin^2/ sin cos)

=( cos-sin)^2 / sin cos}{ (cos+sin)/ sin cos}

= ( cos^2+sin^2-2sin cos)/sin cos}{ cos/sin cos+ sin / sin cos}

= {(1-2sin cos)/sin cos}( cosec+ sec)

=( sec.cosec-2)(cosec+sec)=R.H.S

Hence proved