If `sintheta+cosectheta=2` then the value of `sin^(10)theta+cosec^(10)theta` is

Options

A. 2
B. `2^(10)`
C. `2^(9)`
D. 10

Answer

A

Explanation

Given, ` sin theta + cosec theta =2`
On squaring both sides of Eq. (i) , we get
`sin^(2) theta + cosec^(2) + 2 =4`
` Rightarrow sin^(2) theta +cosec ^(2) theta =2`
Again, on squaring both sides of Eq. (ii) , we get
` sin^(4) theta + cosec^(4) theta =2 `
on cubing both sides of Eqs. (ii) we get
` sin^(2) + cosec^(2) theta)^(3) =2^(3)`
` Rightarrow sin^(6) theta cosec^(2) theta + 3 sin^(2) theta cosec^(2) theta (sin^(2) theta +cosec^(2) theta ) =8`
` Rightarrow sin^(6) theta + cosec^(6) theta + 3.2 =8`
` Rightarrow sin^(6) theta + cosec^(6) theta =2`
On multiplying Eqs. (iv) and (iii) we get
` (sin^(4) theta + cosec^(4) theta) (sin^(6) theta + cosec^(6) theta) =4`
` Rightarrow sin^(10) + sin^(4) theta cosec^(4) theta`
` (sin^(2) theta + cosec^(2) theta) + cosec^(10) theta =4`
`Rightarrow sin^(10) theta + cosec^(10) theta = 4 -2 =2`