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प्रश्न
Given: tan A = `4/3 , "find" : ("cosec""A")/(cot "A"– sec "A")`
उत्तर
Consider the diagram below :
tan A = `(4)/(3)`
i.e. `"perpendicular"/"base" = (4)/(3) ⇒ "BC"/"AB" = (4)/(3)`
Therefore if length of AB = 3x, length of BC = 4x
Since
AB2 + BC2 = AC2 ... [ Using Pythagoras Theorem ]
( 3x )2 + (4x)2 = AC2
AC2 = 9x2 + 16x2 = 25x2
AC = 5x ...( hypotenuse )
Now
sec A = `" hypotenuse "/"base" = "AC"/"AB" =(5x)/ (3x) = (5)/(3)`
cot A = `" base "/"perpendicular" = "AB"/"BC" = (3x)/ (4x) = (3)/(4)`
cosec A = `" hypotenuse "/"perpendicular" = "AC"/"BC" = (5x)/(4x) = (5)/(4)`
Therefore
`("cosec""A")/(cot"A" – sec "A")`
= `(5/ 4) /(3/4 – 5/3)`
= ` (5 /4)/(– 11/12)`
= `– (60)/(44) `
= `– (15)/(11)`
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