State True or False.
The assertion $t a n θ 1 + cot θ + c o t θ 1 - tan θ = 1 + s e c θ ⋅ c o s e c θ$ , is correct for all acute angles.

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True

False

answer is A.

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Detailed Solution

Given equation is

t a n θ 1 + cot θ + c o t θ 1 - tan θ = 1 + s e c θ ⋅ c o s e c θ

.
We start with LHS,

tan θ 1 − c o t θ + c o t θ 1 − tan θ ⇒ sin θ cos θ 1 − cos θ sin θ + cos θ sin θ 1 − sin θ cos θ

⇒ sin θ × sin θ cos θ sin θ − cos θ + cos θ × cos θ sin θ cos θ − sin θ ⇒ sin 2 θ cos θ sin θ − cos θ − cos 2 θ sin θ sin θ − cos θ

Taking the LCM of denominators,

⇒ sin 3 θ − cos 3 θ cos θ sin θ sin θ − cos θ

Using the identity

a 3 − b 3 = a − b a 2 + a b + b 2

, we get

⇒ sin θ − cos θ sin 2 θ + cos 2 θ + sin θ cos θ cos θ sin θ sin θ − cos θ ⇒ sin 2 θ + cos 2 θ + sin θ cos θ cos θ sin θ

We know the identity

cos 2 θ + sin 2 θ = 1

⇒ sin θ cos θ + 1 cos θ sin θ ⇒ sin θ cos θ cos θ sin θ + 1 cos θ sin θ ⇒ 1 + 1 cos θ 1 sin θ ⇒ 1 + s e c θ ⋅ cos e c θ = R H S

Since LHS = RHS, the assertion is true i.e.,

t a n θ 1 + cot θ + c o t θ 1 - tan θ = 1 + s e c θ ⋅ c o s e c θ

, is correct for all acute angles.
Therefore, the correct option is 1.

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