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If $θ . (1 + cosθ) (1 - cosθ) =$ λ then the value of λ is

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θ

1

-1

answer is C.

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Detailed Solution

We are given that the equation as
⇒

θ . (1 + cosθ) (1 - cosθ) =

λ
We know that the formula of difference of squares of two numbers as

a^{2} - b^{2} = (a + b) (a - b)

By using the above formula in the given equation then we get
⇒

θ . (1^{2} - {(cosθ)}^{2} =

λ
⇒

θ . (1 - θ) =

λ
We know that the standard identity of the trigonometric ratios that is

\cos^{2} θ + \sin^{2} θ = 1

By using this standard identity in the above equation then we get
⇒

θ

((

\cos^{2} θ + \sin^{2} θ

)−

θ

)=λ
⇒

θ

(

\sin^{2} θ

)=λ
We know that the relation between the sine ratio and cosecant ratio is given as
sinθ=

\frac{1}{cscθ}

By using this relation in above equation then we get
⇒

θ

(

\frac{1}{θ}

)=λ
⇒λ=1
Therefore we can conclude that the value of λ from the given equation is 1
So, option (3) is the correct answer.

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