यदि `sec theta =x+(1)/(4x)` तो, सिद्ध करो कि ` sec theta+ tan thea =2x` अथवा `sec theta+ tan theta =(1)/(2x)`

Explanation

दिया है:
`sec theta=x+(1)/(4x)`
`therefore tan ^(2)theta =sec ^(2) -1`
`implies tan ^(2)theta (x+(1)/(4x))-1`
`impliestan ^(2) theta=x^(2) +(1)/(16x^(2))+1/2-1`
`impliestan ^(2)theta =x^(2) +(1)/( 16x^(2))-1/2`
`impliestan ^(2) theta =(x- (1)/(4x))^(2)`
`implies tan theta =pm (x-(1)/(4x))`
`implies tan theta = (x-(1)/(4x))` अथवा, `tan theta = -(x-(1)/(4x))`
यदि `tan theta=- (x-(1)/(4x))` है , तो
`sec theta+ tan theta= x+(1)/(4x)+x-(1)/(4x)=2x`
यदि `tan theta =-(x-(1)/(4x))` है, तो
`sec theta + tan theta=(x+(1)/(4x))-(x-(1)/(4x))=(2)/(4x)=(1)/(2x)`
अतः अतः `sec theta+ tan theta =2x `
`sec theta +tan theta =(1)/(2x)`
वैकल्पिक विधि: माना `sec theta +tan theta =lamda " "...(i)`
`therefore sec ^(2)theta - tan ^(2)theta =1`
`implies(sec theta -tan theta ) (sec theta +tan theta)=1`
`implieslamda(sec theta -tan theta)=1" "`(i) से
`implies sec theta= tan theta =(1)/(lamda)" "...(ii)`
सभी (i ) तथा (ii ) को जोड़ने पर
`2 sec theta =lamda + (1)/(lamda)`
`implies2(x+(1)/(4x))=lamda +(1)/(lamda)" "[because sec theta =x+(1)/(4x)("दिया है")]`
`implies2x+(1)/(2x) =lamda+ (1)/(lamda)`
`implies lamda=2x` या `lamda=(1)/(2x)" "` [दोनों पक्षों कि तुलना करने पर]
`implies sec theta+ tan theta =2x, (1)/(2x)`