Question

prove that : cot³0/1+cot²0+tan³0/1+tan²0=sec0. cosec0-2sin0. cos0

Answer 1

Ans. Taking LHS,
cot³0/1+cot²0+tan³0/1+tan²0
= cot³0/cosec²0+ tan³0/sec²0
= cos³0/sin³0 / 1/sin²0+ sin³0/cos³0 / 1/cos²0
= cos³0/sin³0×sin²0+ sin³0/cos³0×cos²0
= cos³0/sin0+sin³0/cos0
=sin⁴0+cos⁴0/sin0.cos0
= (sin²0+cos²0)² - 2sin²0cos²0/ sin0.cos0
= 1-2sin²0.cos²0/sin0.cos0
= 1/sin0.cos0-2sin0.cos0
= sec0.cosec0-2sin0.cos0
= RHS

Answer 2

Ans. Taking LHS,

cot³0/1+cot²0+tan³0/1+tan²0

= cot³0/cosec²0+ tan³0/sec²0

= cos³0/sin³0 / 1/sin²0+ sin³0/cos³0 / 1/cos²0

= cos³0/sin³0×sin²0+ sin³0/cos³0×cos²0

= cos³0/sin0+sin³0/cos0

=sin⁴0+cos⁴0/sin0.cos0

= (sin²0+cos²0)² - 2sin²0cos²0/ sin0.cos0

= 1-2sin²0.cos²0/sin0.cos0

= 1/sin0.cos0-2sin0.cos0

= sec0.cosec0-2sin0.cos0

= RHS

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