Simplify (1 – cos θ) (1 + cos θ) (1 + cot2 θ)

1.	Simplify (1 – cos θ) (1 + cos θ) (1 + cot2 θ)
Answer» Given that (1 – cos θ) (1 + cos θ) (1 + cot² θ) = (1 – cos² θ) (1 + cot² θ) [∵ (a – b) (a + b) = a² – b²] = sin² θ. cosec² θ [∵ 1 – cos² θ = sin² θ; 1 + cot² θ = cosec² θ] = sin² θ . \(\frac{1}{sin^2\, θ}\) [∵ cosec θ = \(\frac{1}{sin\, θ}\) ] = 1

Answer»

Given that

(1 – cos θ) (1 + cos θ) (1 + cot² θ)

= (1 – cos² θ) (1 + cot² θ)

[∵ (a – b) (a + b) = a² – b²]

= sin² θ. cosec² θ [∵ 1 – cos² θ = sin² θ; 1 + cot² θ = cosec² θ]

= sin² θ . \(\frac{1}{sin^2\, θ}\) [∵ cosec θ = \(\frac{1}{sin\, θ}\) ]

= 1

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