A is an angle in the fourth quadrant. It satisfies the trigonometric equation `3 (3 - tan^2 A-cot A)^2 = 1.` Which one of the following is a value of A?

Options

A. `300^(@)`
B. `315^(@)`
C. `330^(@)`
D. `345^(@)`

Answer

A

Explanation

Checking through options
`300^(@)=-60^(@)`
So , `3[3-tan^(2)(-60^(@))-cot(-60^(@))]^(2)`
`=3[3-3+(1)/(sqrt(3))]^(2)=3xx(1)/(3)=1`
(b) Given equation :
`"cosec"x+cotx=sqrt(3)`
`"cosec"x-cotx=(1)/(sqrt(3))`
`rArr2" cosec "x=sqrt(3)+(1)/(sqrt(3))=(4)/(sqrt(3))`
`rArr"cosec "x=(2)/(sqrt(3))`
`rArrsinx=(sqrt(3))/(2)`
Possible values of x `=(pi)/(3),(2pi)/(3)`
`therefore` Option (b) is correct . (b)
`[(2costheta+1)(2costheta-1)]^(10)[2cos2theta-1]^(10)[2cos4theta-1]^(10)`
when , `theta=(pi)/(8)` (given)
`=(4"cos"^(2)(pi)/(8)-1)^(10)(2xx(1)/(sqrt(2))-1)^(10)(-1)^(10)`
`=(4xx(2+sqrt(2))/(4)-1)^(10)(sqrt(2)-1)^(10)`
`=[(sqrt(2)+1)(sqrt(2)-1)]^(10)=1^(10)=1`
(a) Product pf roots `=(c)/(a)`
`rArrcosalpha.cosbeta=-(3)/(4)`
`rArr(1)/(cosalpha.cosbeta)=secalpha.secbeta=-(4)/(3)`
(b) `A=sin^(2)theta+cos^(4)theta`
`=sin^(2)theta+(1-sin^(2)theta)^(2)`
`=1+sin^(4)theta-sin^(2)theta`
`=1-sin^(2)theta(1-sin^(2)theta)`
`=1-sin^(2)theta.cos^(2)theta`
`=(4-4sin^(2)theta.cos^(2)theta)/(4)=(4-sin^(2)(2theta))/(4)`
As , we know , `0lesin^(2)2thetale1`
`thereforeA=(4-0)/(4)or(4-1)/(4)rArr(3)/(4)leAle1`
(d) `25" cosec"^(2)x+36sec^(2)x`.
Minimum value `=(sqrt(25)+sqrt(36))^(2)`
`=(5+6)^(2)=(11)^(2)=121`