यदि `cos^(-1)""(p)/(a)+cos^(-1)""(q)/(b)=alpha ` , सिद्ध कीजिए कि
` (p^(2))/(a^(2))-(2pq)/(ab)cos alpha+(q^(2))/(b^(2))=sin^(2) alpha`

Explanation

मान `cos^(-1)""(p)/(a)=A, cos^(-1)""(q)/(b)=B`
`implies cosA =(p)/(a), cos B=(q)/(b)`
`:. A+B=alpha `
`implies cos (A+B)=cos alpha `
`cosA cos B-sinA sinB= cos alpha `
` cos A * cosB- cos alpha=sinA sinB`
`implies (p)/(a) xx (q)/(b)- cos alpha=sqrt(1-(p^(2))/(a^(2)))sqrt(1-(q^(2))/(b^(2)))`
दोनों ओर का वर्ग करने पर
` ((pq)/(ab))^(2)+cos^(2) alpha-(2pq)/(ab)cos alpha=(1-(p^(2))/(a^(2)))(1-(q^(2))/(b^(2)))`
`implies (p^(2)q^(2))/(a^(2)b^(2))+ cos^(2)alpha-(2pq)/(ab) cos alpha=1-(q^(2))/(b^(2))-(p^(2))/(a^(2))+(p^(2)q^(2))/(a^(2)b^(2))`
`implies (p^(2))/(a^(2))-(2pq)/(ab)cos alpha+(q^(2))/(b^(2))=1-cos^(2)alpha`
`implies (p^(2))/(a^(2))-(2pq)/(ab)cos alpha +(q^(2))/(b^(2))=sin^(2)alpha `