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The value of ^{ 2 }{ ( ^{ -1 }{ 2 } ) } +{cosec} ^{ 2 }{ (

Question

The value of $sec^{2} (tan^{- 1} 2) + cosec^{2} (cot^{- 1} 3) =$ ____

6

15

13

25

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The value of

sec^{2} (tan^{- 1} 2) + cosec^{2} (cot^{- 1} 3) =

____

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Text solutionVerified

Let

tan^{- 1} 2 = x

(1)

2 = tan x

Let

p er p e n d i c u l a r = p = 2 k

ba se = b = k

h y p o t e n u se = h = (2 k)^{2} + k^{2} = (5 k)^{2} = 5 k

sec x = \frac{h y p o t e n u se}{ba se} = \frac{5 k}{k} = 5

x = se c^{- 1} 5

(2)

Let

cot^{- 1} 3 = y

(3)

3 = cot y

Let

ba se = 3 = k^{^{'}}

p er p e n d i c u l a r = p = k^{^{'}}

h y p o t e n u se = h = (k^{^{'}})^{2} + (3 k^{^{'}})^{2} = (10 k^{^{'}})^{2} = 10 k^{^{'}}

cosec y = \frac{h y p o t e n u se}{p er p e n d i c u l a r} = \frac{10 k ^{^{'}}}{k ^{^{'}}} = 10

y = cosec^{- 1} 10

(4)

From (1),(2),(3) and (4)

sec^{2} (tan^{- 1} 2) + cosec^{2} (cot^{- 1} 3) = sec^{2} x + cosec^{2} y = sec^{2} se c^{- 1} 5 + cosec^{2} cosec^{- 1} 10

= (sec sec^{- 1} 5)^{2} + (cosec cosec^{- 1} 10)^{2} = (5)^{2} + (10)^{2} = 5 + 10 = 15

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