Question

prove that cosec A (1-cosA) (cosec A +cotA) =1​

Answer 1

Answer:-

We have to prove:

cosec A (1 - cos A) (cosec A + cot A) = 1

using cosec A = 1/sin A and cot A = cos A/sin A we get,

⟹ (1/sin A) (1 - cos A) (1/sin A + cos A/sin A) = 1

⟹ (1/sin A) (1 - cos A) (1 + cos A) / sin A = 1

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

⟹ (1/sin² A) (1² - cos² A) = 1

⟹ (1/sin² A) (1 - cos² A) = 1

using the identity 1 - cos² A = sin² A we get,

⟹ (1/sin² A) * (sin² A) = 1

⟹ 1 = 1

Hence, Proved.

_________________________________

Another Method:-

We know, cosec A = 1/sin A.

So, first multiply cosec A & 1 - cos A.

⟹ (cosec A - cos A/sin A) (cosec A + cot A) = 1

• cos A/sin A = cot A.

⟹ (cosec A - cot A)(cosec A + cot A) = 1

⟹ cosec² A - cot² A = 1

⟹ 1 = 1 [∵ cosec² θ - cot² θ = 1 ].

Answer 2

Step-by-step explanation:

LHS:-

$\\ \tt{:}\longrightarrow cosecA(1-cosA)(cosecA+cotA)$

$\\ \tt{:}\longrightarrow (cosecA-cosecA\times cosA)(cosecA+cotA)$

$\\ \tt{:}\longrightarrow (cosecA-\dfrac{1}{sinA}\times cosA)(cosecA+cotA)$

$\\ \tt{:}\longrightarrow (cosecA-\dfrac{cosA}{sinA})(cosecA+cotA)$

$\\ \tt{:}\longrightarrow (cosecA-cotA)(cosecA+cotA)$

$\\ \tt{:}\longrightarrow cosec^2A-cot^2A$

$\\ \tt{:}\longrightarrow 1$

$\\ \tt{:}\longrightarrow LHS=RHS\:\:(Proved)$