Question

If tan theta =1/√5 , what is the value of cosec^2theta -sec^2 theta /cosec^2theta +sec^2theta

Answer 1

Answer: 12/13

Step-by-step explanation:

Since, tan theta = 1/√5 ..(1)

We know that,

tan theta = perpendicular / base

So from (1) , perpendicular = 1x

base = √5 x

{Here,x is the constant by which both numerator and denominator given above is divided.}

Let the hypotenuse be K. Then,

By Pythagoras Theorem,

(Hypotenuse)^2 = (Base)^2 +

(Perpendicular)^2

K^2 = (1x)^2 + (√5 x)^2

K^2 = x^2 + 5x^2

K^2 = 6x^2

K = √6 x

Hence, Hypotenuse = K= √6 x

Here,

cosec theta = hypotenuse/perpendicular = √6 x /x

= √6/1

sec theta = hypotenuse/base

= √6 x/ √5 x

√6/√5

NOW, According To The Question,

{(cosec theta)^2 - (sec theta)^2} / {(cosec theta)^2 + (sec theta)^2} =

= {(√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

= 2/3

Hence,the required answer = 2/3