Q. 2 4.4( 159 Votes )

Evaluate



OR

If cosec2 θ (1 + cos θ) (1 – cos θ) = k, then find the value of k.

Answer :

Given:


[, tan θ = cot (90° - θ)]



= 1


OR


Given: cosec2 θ (1 + cos θ) (1 – cos θ) = k


To find: Value of k


Proof: Taking LHS


= cosec2 θ (1 + cos θ) (1 – cos θ)


Using the identity,


(a + b)(a – b) = (a2 – b2)


= cosec2 θ [(1)2 – (cosθ)2]


= cosec2 θ [ 1 – cos2θ] …(i)


We know that,


cos2θ + sin2θ = 1


or sin2θ = 1 – cos2θ


So, equation (i) becomes,


= cosec2θ (sin2θ) …(ii)


Now, we also know that,



So, equation (ii) becomes



= 1


cosec2 θ (1 + cos θ) (1 – cos θ) = 1 = k


Hence, the value of k is 1


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