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Write LHS in terms of sine and cos.

LHS

= sin² / [cos (sin+cos)] + cos² / [sine (cos - sine)]

= [ Sin³ (cos - sine) + Cos³ (sin + cos) ] / [sine cos (cos² - sin²)]

= [ Sin cos (cos²+sin²) - (cos² - sin²)(cos²+sin²) ] / [sine cos (cos2 ) ]

= [ sin cos - cos2 ] / [sine cos cos2]

= Sec2θ - 2 cosec2θ

this is the answer for LHS.

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the two expressions on the right side are equal.

1+ tan + cot = 1 + (sin²+ cos²)/(sin cos ) = 1 + cosec sec.

LHS

= sin² / [cos (sin+cos)] + cos² / [sine (cos - sine)]

= [ Sin³ (cos - sine) + Cos³ (sin + cos) ] / [sine cos (cos² - sin²)]

= [ Sin cos (cos²+sin²) - (cos² - sin²)(cos²+sin²) ] / [sine cos (cos2 ) ]

= [ sin cos - cos2 ] / [sine cos cos2]

= Sec2θ - 2 cosec2θ

this is the answer for LHS.

==========================

the two expressions on the right side are equal.

1+ tan + cot = 1 + (sin²+ cos²)/(sin cos ) = 1 + cosec sec.