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What is the value of x of the equation $cos (270 ∘ + θ) + x ⋅ cos θ ⋅ cot (180 ∘ + θ) = sin (270 ∘ − θ)$ ?

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− tan 2 θ + tan θ

− tan 2 θ − tan θ

tan 2 θ + tan θ

tan 2 θ − tan θ

answer is B.

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Detailed Solution

Let the given be,

cos (270 ∘ + θ) + x ⋅ cos θ ⋅ cot (180 ∘ + θ) = sin (270 ∘ − θ)

… (1)
We know that,

cot (180 ∘ + θ) = cos (180 ∘ + θ) sin (180 ∘ + θ) = − cos θ − sin θ = cos θ sin θ

cos (270 ∘ + θ) = cos (180 ∘ + 90 ∘ + θ) = cos (180 ∘ + (90 ∘ + θ)) = − cos (90 ∘ + θ) = sin θ

By using the above two results, L.H.S in the equation becomes,

= sin θ + x ⋅ cos θ ⋅ cos θ sin θ = sin θ + x ⋅ cos 2 θ sin θ = sin 2 θ + x ⋅ cos 2 θ sin θ

Now, R.H.S in the equation becomes,

sin (270 ∘ − θ) = sin (180 ∘ + 90 ∘ − θ) = sin (180 ∘ + (90 ∘ − θ)) = − sin (90 ∘ − θ) = − cos θ

Then, the equation becomes,

sin 2 θ + x ⋅ cos 2 θ sin θ = − cos θ sin 2 θ + x ⋅ cos 2 θ = − cos θ sin θ x ⋅ cos 2 θ = − sin 2 θ − cos θ sin θ x = − sin 2 θ cos 2 θ − cos θ sin θ cos 2 θ x = − tan 2 θ − tan θ

Hence, the correct option is 2.

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