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Prove That: (Sin θ + 1 + Cos θ) (Sin θ − 1 + Cos θ) . Sec θ Cosec θ = 2 - Mathematics

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Sum

Prove that:
(sin θ + 1 + cos θ) (sin θ − 1 + cos θ) . sec θ cosec θ = 2

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Solution

LHS = (sinθ + 1 + cosθ)(sinθ − 1 + cosθ). secθcosecθ

= [sin2θ − sinθ + sinθcosθ + sinθ − 1 + cosθ + sinθcosθ − cosθ + cos2θ] `1/cosθ1/sinθ `                                                             ...(∵ secθ = `1/cosθ  and  cosecθ = 1/sinθ`)

= [1 + 2sinθcosθ − 1]`1/cosθ  1/sinθ`

= [2sinθcosθ]`1/cosθ1/sinθ`

= 2 = RHS             

Hence proved.

Concept: Trigonometric Ratios of Complementary Angles
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