uired ANSWER:-

Given:

• tan²θ(cosec θ - 1)(cosec θ + 1) = k

To Find:

• The value of k.

Solution:

Given,

→ tan²θ(cosec θ - 1)(cosec θ + 1) = k

Using identity (a + b)(a - b) = a² - b², we get,

→ tan²θ × (cosec²θ - 1) = k

We know that,

→ cosec²θ - cot²θ = 1

→ cosec²θ - 1 = cot²θ

Substituting the value in the equation, we get,

→ tan²θ × cot²θ = k

As tan θ is the reverse of cot θ i.e.,

→ tan θ = 1/cot θ

→ 1/cot²θ × cot²θ = k

→ 1 = k

→ k = 1

→ So, the value of k is 1.

Answer:

• k = 1

Additional Formulae:

1. RELATIONSHIP between sides and T-Ratios.

• sin θ = Height/Hypotenuse.
• cos θ = Base/Hypotenuse.
• tan θ = Height/Base.
• cot θ = Base/Height.
• SEC θ = Hypotenuse/Base.
• cosec θ = Hypotenuse/Height.

2. RECIPROCAL Identities.

• sin θ = 1/cosec θ and cosec θ = 1/sin θ
• cos θ = 1/sec θ and sec θ = 1/cot θ
• tan θ = 1/cot θ and cot θ = 1/tan θ

3. Co-function identities.

• sin(90° - θ) = cos θ and cos(90° - θ) = sin θ.
• tan(90° - θ) = cot θ and cot(90° - θ) = tan θ.
• sec(90° - θ) = cosec θ and cosec(90° - θ) = sec θ

4. Pythagoras identities.

• sin²θ + cos²θ = 1
• cosec²θ - cot²θ = 1
• sec²θ - tan²θ = 1