Question

(1-tanx)square + ( 1 - cotx)square = ( secx - cosec x )

Answer 1

Answer:

( 1 - Tan x )² + ( 1 - Cot x )² = ( Sec X - Cosec X )²

LHS:

=> ( 1 + Tan² X + 2 Tan X ) + ( 1 + Cot² X - 2 Cot x )

=> 1 + Tan² X + 2 Tan X + 1 + Cot² X - 2 Cot X

=> Sec² X + Cosec² X + 2 Tan X - 2 Cot X

=> Sec² X  - 2 ( Tan X + Cot X ) + Cosec² X

=> Sec² X - 2 ( Sin X / Cos X + Cos X / Sin X ) + Cosec² X

=> Sec² X - 2 ( Sin² X + Cos² X / Sin X . Cos X ) + Cosec² X

=> Sec² X - 2 ( 1 / Sin X . Cos X ) + Cosec² X

=> Sec² X - 2 ( Sec X . Cosec X ) + Cosec² X

=> ( Sec X - Cosec X )²

Hence Proved !