यदि `"cos"^(-1)(x)/(a)+"cos"^(-1)(y)/(b)=alpha` तब सिद्ध कीजिए - `(x^(2))/(a^(2))-(2xy)/(ab)cos alpha + (y^(2))/(b^(2))=sin^(2)alpha`.

Explanation

यहाँ
`"cos"^(-1)(x)/(a)+"cos"^(-1)(y)/(b)=alpha`
`rArr cos^(-1)[(x)/(a).(y)/(b)-sqrt(1-(x^(2))/(a^(2)))sqrt(1-(y^(2))/(b^(2)))]=alpha`
`rArr cos^(-1)[(xy)/(ab)-sqrt(1-(x^(2))/(a^(2)))sqrt(1-(y^(2))/(b^(2)))]=alpha`
`rArr " " (xy)/(ab)-sqrt(1-(x^(2))/(a^(2)))sqrt(1-(y^(2))/(b^(2)))=cos alpha`
`rArr ((xy)/(ab)-cos alpha)= sqrt(1-(x^(2))/(a^(2)))sqrt(1-(y^(2))/(b^(2)))`
दोनों पक्षों का वर्ग करने पर,
`((xy)/(ab)-cos alpha)^(2)= (1-(x^(2))/(a^(2)))(1-(y^(2))/(b^(2)))`
`rArr (x^(2)y^(2))/(a^(2)b^(2))-(2xy)/(ab)cos alpha + cos^(2)alpha`
`= 1-(x^(2))/(a^(2))-(y^(2))/(b^(2))+(x^(2)y^(2))/(a^(2)b^(2))`
`rArr (x^(2))/(a^(2))+(y^(2))/(b^(2))-(2xy)/(ab)cos alpha = 1- cos^(2)alpha`
`rArr (x^(2))/(a^(2))+(y^(2))/(b^(2))-(2xy)/(ab)cos alpha = sin^(2)alpha`.
यही सिद्ध करना था |