Question

if tan theta equals 1/root 3 find the value of cosec 2 theta-sec 2 theta by cosec 2 theta+sec 2 theta

Answer 1

Let theta = A.

As we know that,

1 ) tan A = 1/Cot A

2 ) cosec² A = 1 + cot² A

3 ) sec² A = 1 + tan² A

Given, tan A = 1/√3 ---- eq. (1)

or cot A = √3 ------ eq.(2)

Now, (cosec² A - sec² A ) = 1 + cot² A - (1 + tan² A )

Or, 1 + cot² A - 1 - tan² A = cot² A - tan² A

By putting the value of tan A and cot A from eq. (1) and (2), we get  

= ( √3 )² - ( 1/√3 )²

= 3 - 1/3

= ( 9 - 1 ) / 3

(cosec² A - sec² A ) = 8/3 --- eq. (3)

Also, cosec² A + sec² A= 1 + cot² A + 1 + tan² A

Or, 2 + cot² A + tan² A

By putting the value of tan A and cot A from eq.(1) and(2), we get  

= 2 + ( √3 )² + ( 1/√3 )²

= 2 + 3 + 1/3

= 5 + 1/3

= ( 15 + 1 )/3

cosec² A + sec² A = 16/3 ---- eq. (4)

Therefore, dividing eq. (3) by eq. (4) we get the required value, i.e.  

(cosec² A - sec² A)/ cosec² A + sec² A= ( 8/3 ) / ( 16/3 )

=> 8/16

=> 1/2