If tan theta = 1/root 5 find the value of cosec square theta -sec square theta /cosec square theta + sec square theta
Answers
Answer:
{ ( cosec )^2 theta - ( sec )^2 theta } /
{ ( cosec )^2 theta - ( sec )^2 theta} =
= 2/3
Step-by-step explanation:
Given,
tan theta = 1/√5 ...(1)
We know that,
tan theta = perpendicular/ base ..(2)
From eq. (1) and eq.(2) , we get
perpendicular / base = 1/√5
So,
Perpendicular = 1 x
Base = x√5
Here 'x' is the constant by which both 1 and √5 are multiplied.
Since the given triangle is right angled triangle, then
By Pythagoras Theorem, we get
(Hypotenuse)^2 = (Perpendicular)^2
+ (Base)^2
(Hypotenuse)^2 = (1 x)^2 + (x√5)^2
=> (Hypotenuse)^2 = 1 x^2 + 5 x^2
=> (Hypotenuse)^2 = 6 x^2
=> Hypotenuse = √(6 x^2)
=> Hypotenuse = x√6
Hence, Hypotenuse = x√6
We know that,
cosec theta =
= Hypotenuse/Perpendicular
= x√6 / 1 x = √6/1
sec theta =
= Hypotenuse / Base
= x√6 / x√5 = √6 / √5
Now according to the question :
{ (cosec)^2 theta - (sec)^2 theta } /
{ (cosec)^2 theta + (sec)^2 theta } =
= {(√6/1)^2 - (√6/√5)^2} /
{(√6/1)^2 + (√6/√5)^2}
= {(6/1) - (6/5)} / {(6/1) + (6/5)}
= { ( 30 - 6 ) / 5 } / { ( 30 + 6 ) / 5 }
= { 24 / 5 } / { 36 / 5 }
= 24 / 36
= 12 / 18
= 2 / 3
Hence,the required answer = 2/3
Answer:
Step-by-step explanation:
Answer:
{ ( cosec )^2 theta - ( sec )^2 theta } /
{ ( cosec )^2 theta - ( sec )^2 theta} =
= 2/3
Step-by-step explanation:
Given,
tan theta = 1/√5 ...(1)
We know that,
tan theta = perpendicular/ base ..(2)
From eq. (1) and eq.(2) , we get
perpendicular / base = 1/√5
So,
Perpendicular = 1 x
Base = x√5
Here 'x' is the constant by which both 1 and √5 are multiplied.
Since the given triangle is right angled triangle, then
By Pythagoras Theorem, we get
(Hypotenuse)^2 = (Perpendicular)^2
+ (Base)^2
(Hypotenuse)^2 = (1 x)^2 + (x√5)^2
=> (Hypotenuse)^2 = 1 x^2 + 5 x^2
=> (Hypotenuse)^2 = 6 x^2
=> Hypotenuse = √(6 x^2)
=> Hypotenuse = x√6
Hence, Hypotenuse = x√6
We know that,
cosec theta =
= Hypotenuse/Perpendicular
= x√6 / 1 x = √6/1
sec theta =
= Hypotenuse / Base
= x√6 / x√5 = √6 / √5
Now according to the question :
{ (cosec)^2 theta - (sec)^2 theta } /
{ (cosec)^2 theta + (sec)^2 theta } =
= {(√6/1)^2 - (√6/√5)^2} /
{(√6/1)^2 + (√6/√5)^2}
= {(6/1) - (6/5)} / {(6/1) + (6/5)}
= { ( 30 - 6 ) / 5 } / { ( 30 + 6 ) / 5 }
= { 24 / 5 } / { 36 / 5 }
= 24 / 36
= 12 / 18
= 2 / 3