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If 4costheta-3sectheta=2tantheta, then ...

If `4costheta-3sectheta=2tantheta`, then `theta` is equal to

A

`npi+(-1)^(n)(pi)/(10)`

B

`npi+(-1)^(n)(pi)/(6)`

C

`npi+(-1)^(n)(3pi)/(10)`

D

`npi`

Text Solution

Verified by Experts

The correct Answer is:
A
Since ,`4costheta-3sectheta=2tantheta`
`rArr4costheta-(3)/(costheta)=2(sintheta)/(costheta)`
`rArr4cos^(2)theta-3=2sintheta`
`rArr4cos^(2)theta-3=2sintheta`
`rArr4-4sin^(2)theta-3=2sintheta`
`rArr4sin^(2)theta+2sintheta-1=0`
`rArrsintheta=(-2+-sqrt(4+16))/(8)`
`rArrsintheta=(-+-sqrt(5))/(4)`
Either `sintheta=(-1+sqrt(5))/(4)orsintheta=(-1-sqrt(5))/(4)`
`rArrsintheta="sin"(pi)/(10)orsintheta=sin(-(3pi)/(10))`
`[because"sin"(pi)/(10)=(sqrt(5)-1)/(4),"sin"(3pi)/(10)=(sqrt(5)+1)/(4)]`
`rArrtheta=npi+(-1)^(n)(pi)/(10)`
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