यदि `sec (theta - alpha), sec theta` और `sec(theta + alpha)`,समांतर श्रेणी `(AP)` में हैं , जहाँ है `cosalpha ne 1`, तो `sin^(2) theta + cos alpha` के मान क्या है ?

Options

A. 0
B. 1
C. -1
D. `1/2`

Explanation

यदि `sec (theta - alpha), sec theta` और `sec(theta + alpha)`
`2 sec theta = sec (theta - alpha) + sec( theta + alpha)`
`rArr (2)/(cos theta) = (1)/(cos(theta - alpha)) + (1)/(cos(theta + alpha))`
`rArr (2)/(cos theta) = (cos(theta + alpha)+cos(theta - alpha))/(cos (theta - alpha)(cos theta + alpha))`
`rArr 2 cos ((theta + alpha + theta - alpha)/(2)) cos`
`2/(cos theta) = (((theta + alpha + theta - alpha)/(2)))/(cos^(2) theta - sin^(2) alpha)`
`[{:( :' cos C + cos D = 2 cos((C+D)/(2))),( "तथा" cos ((C-D)/(2))):}]`
`cos (A + B) cos (A-B) = cos^(2) A - sin^(2)B`
`rArr (2)/(cos theta) = (2 cos theta cos alpha)/(cos^(2) theta - sin^(2) alpha)`
`rArr cos^(2) theta cos alpha = cos^(2) - sin^(2) alpha`
`rArr sin^(2) alpha = cos^(2) theta - cos^(2) theta cos alpha`
`rArr sin^(2) alpha = cos^(2) theta ( 1- cos alpha)`
`rArr cos^(2) theta = (sin^(2)alpha)/(1-cos alpha)`
`rArr cos^(2) theta = ((1+cosalpha) (1-cos alpha))/(1-cos alpha)`
`rArr cos^(2) theta = 1 + cos alpha`
`rArr 1- sin^(2) theta = 1 + cos alpha`
`[ :' sin^(2) theta + cos^(2) theta = 1]`
`rArr sin^(2) theta + cos alpha = 0`