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Prove that cosec(^{-1}((^{-1}((^{-1}a))))) = {3 - a^2}, where

Question

Prove that $cosec (tan^{- 1} (cos (cot^{- 1} (sec (sin^{- 1} a))))) = 3 - a^{2}$ , where $a \in [0, 1]$ .

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Samuel Discussed

Prove that

cosec (tan^{- 1} (cos (cot^{- 1} (sec (sin^{- 1} a))))) = 3 - a^{2}

, where

a \in [0, 1]

7 mins ago

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Text solutionVerified

given,

cosec (tan^{- 1} (cos (cot^{- 1} (sec (sin^{- 1} a)))))

= cosec (tan^{- 1} (cos (cot^{- 1} (sec (sec^{- 1} \frac{1}{1 - a ^{2}})))))

{Since for

a \in [0, 1], sin^{- 1} a = sec^{- 1} \frac{1}{1 - a ^{2}}

]

= cosec (tan^{- 1} (cos (cot^{- 1} \frac{1}{1 - a ^{2}})))

= cosec (tan^{- 1} (cos (cos^{- 1} \frac{1}{2 - a ^{2}})))

[Since for

a \in [0, 1], cot^{- 1} \frac{1}{1 - a ^{2}} = cos^{- 1} \frac{1}{2 - a ^{2}}

]

= cosec (tan^{- 1} \frac{1}{2 - a ^{2}})

= cosec (cose c^{- 1} 3 - a^{2})

[Since for

a \in [0, 1], tan^{- 1} \frac{1}{2 - a ^{2}} = cose c^{- 1} 3 - a^{2}

]

= 3 - a^{2}

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Question Text	Prove that $cosec (tan^{- 1} (cos (cot^{- 1} (sec (sin^{- 1} a))))) = 3 - a^{2}$ , where $a \in [0, 1]$ .
Answer Type	Text solution:1
Upvotes	150