Question

(1 + cot2 theta) (1 - cos theta) (1 + cos theta) = 1

Answer 1

(1 + cot2∅)(1 - cos∅)(1 + cos∅) = 1

{use , (a + b)(a - b) = a² - b² }

(1 + cot2∅)( 1 - cos²∅) = 1

[ use, 1 - cos²x = sin²x]

(1 + cot2∅)(sin²∅) = 1

( 1 + tan2∅)(sin²∅) = tan2∅

use, formula,
tan2∅ = 2tan∅/(1 - tan²∅)

( 1 - tan²∅ +2tan∅)sin²∅/(1 - tan²∅) = 2tan∅/( 1 - tan²∅)

(1 -tan²∅+ 2tan∅)sin²∅ = 2tan∅

( 1 - tan²∅+2tan∅)/cosec²∅ = 2tan∅

( 1 - tan²∅ + 2tan∅)/(1 + cot²∅) = 2tan∅

( 1 - tan²∅ + 2tan∅)tan∅/(1 +tan²∅) = 2

tan∅ - tan³∅ +2tan²∅ = 2 + 2 tan²∅

tan³∅ - tan∅ +2 = 0

Answer 2

Hey there !!!

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(1+cot²θ)(1-cosθ)(1+cosθ)=1--------Equation 1

(1+cosθ)(1-cosθ) is of the form (a+b)(a-b)=a²-b²

(1+cosθ)(1-cosθ)=1-cos²θ

But 1-cos²θ=sin²θ  

So equation 1 changes to 

=(1 + cot²θ )(sin²θ)

=sin²θ+cot²θsin²θ

=sin²θ+(cos²θ*sin²θ/sin²θ) 

=sin²θ+cos²θ

=1

  LHS=RHS

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Hope this helped you...................