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prove the identity:sin2A/cos2A + cos2A/sin2A = 1/sin2A.cos2A - 2

To prove: sin^2 A/cos^ 2 A + cos^2 A/sin^2 A + 2 = 1/sin^2 A. cos^2 A LHS= sin^2 A/cos^ 2 A + cos^2 A/sin^2 A + 2 = (sin A/cos A)^2 + (cos A/sin A)^2 + 2 = tan^2 A + cot^2 A + 2 = (tan A + cot A)^2 = (sin A/cos A + cos A/ sin A)^2 = (sin^2 A + cos^2 A/sin A. cos A)^ 2 = (1/sin A. cos A)^ 2 = 1/sin^2 A. cos^2 A = RHS
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LHS = sin2A +  cos2A = tan2A + cot2A
            cos2A     sin2A
        = (sec2A -1) + (cosec2A-1)            [tan2A = sec2A -1  AND  cot2A = cosec2A -1]
        =    1      +    1     - 2                       
            cos2A    sin2A                                 
        = sin2A + cos2A -2                                  
            sin2A.cos2A                                     
        =          1          -2                           [sin2A + cos2A =1]                   
           sin2A.cos2A                                               
        = RHS
Hence Proved.
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To Prove: sin2 A/cos2 A + cos2 A/sin2 A + 2 = 1/sin2 A. cos​2 A

LHS = sin2 A/cos2 A + cos2 A/sin2 A + 2
= (sin A/cos A)2 + (cos A/sin A)2 + 2
= tan2 A + cot2 A + 2
= (tan A + cot A)2
= (sin A/cos A + cos A/sin A)2
= (sin2 A + cos2 A/sin A. cos A)2
= (1/sin A. cos A)2
= 1/sin2 A. cos2 A
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LHS = sin2 A/cos2 A + cos2 A/sin2 A
= sin4 A + cos4 A/ sin2.cos2 A
= (sin2 A)2 + (cos2 A)2/ sin2.cos2 A
= (sin2​ A + cos2 A)2 - 2 sin2 A. cos2 A/ sin2.cos2 A
= 1 - 2 sin2 A. cos2 A/ sin2.cos2 A
= 1/ sin2.cos2 A - 2 sin2 A. cos2 A/ sin2.cos2 A
= 1/ sin2.cos2 A - 2 = RHS
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