Question

prove that
(1+tan²A/1+cot²A)=(1-tan²A/1-cotA)²=tan²A​

Answer 1

Answer:

( 1 + tan²A / 1 + cot²A ) = ( 1 - tan A / 1 - cot A )² = tan²A​

Step-by-step explanation:

LHS

( 1 + tan²A / 1 +cot²A )

= 1 + tan²A / 1 + 1/tan²A

=  1 + tan²A / tan²A + 1 / tan²A

= Tan²A ( 1 + tan²A / tan²A + 1 )

= tan²a

RHS

 ( 1 - tan A / 1 - cot A )²

=  (1 - tan A)² / (1 - cot A)²

= 1 + tan²A - 2tan A / 1 + cot²A - 2cot A

= sec²A - 2tan A / cosec²A - 2cot A

= 1/cos²A - 2sinA/cosA  /  I/sin²A - 2cosA/sinA ( take common denominators)

= 1/cos²A - 2sinAcosA/cos²A  /  I/sin²A - 2cosAsinA/sin²A

=  1 - 2sinAcosA / cos²A  /  I- 2cosAsinA/sin²A

= 1 - 2sinAcosA / cos²A × sin²A/ I- 2cosAsinA

= sin²A / cos²A

= Tan²A

∴ LHS = RHS = tan²A

Hence Proved

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