यदि `sec theta=x+(1)/(4x) (0^(@) lt 0 lt 90^(@))` है तो `sec theta+tan theta` किसके बराबर होगा?)

Options

A. `(x)/(2)`
B. 2x
C. x
D. `(1)/(2x)`

Answer

b

Explanation

`sec theta=(4x^(2)+1)/(4x)`
`tan theta=sqrt(sec^(2)theta-1)`
`=sqrt([(4x^(2)+1)/(4x )]^(2)-1)`
`=sqrt(((4x^(2)+ 1)^(2)-(4x)^(2))/((4x) ^(2)))`
`=sqrt((16x^(4)+ 1+8x^(2)-16x^ (2))/((4x)^(2)))`
`=sqrt((16x^(2)+1- 8x^(2))/((4x )^(2)))`
`=sqrt(((4x^(2)-1)^(2))/((4x)^(2)))`
` =(4x^(2)-1)/(4x)`
`therefore theta+tan theta=(4x^(2)+1)/(4x) +(4x^(2)-1)/(4x)`
`=(4x^(2)+1 +4x^(2)-1)/(4x )=(8x^(2))/(4 x)=2x`
Alternate
`sec theta=x+(1)/(4x)`
Put x =1
`sec theta=1+(1)/(4)=(5)/(4)=(H)/(B)`
`tan theta=(P)/(B)=(3)/(4)`
Now,
`sectheta+tan theta`
`=(5)/(4)+(3)/(4)=(5+3)/(4)=(8)/(4)=2xx1=2x`,
(x=1)