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प्रश्न
If sin θ = `a/b`, show that `(sectheta + tan theta) = sqrt((b+a)/(b-a))`
उत्तर
Given: `sintheta = a/b`
To show: `sec thetatantheta = sqrt((b + a)/(b - a))`
Using Trigonometric identity,
`costheta = sqrt(1 - sin^2 theta)`
`costheta = sqrt(1 - (a/b)^2)`
`costheta = sqrt((b^2 - a^2)/(b^2))`
`costheta = sqrt((b^2 - a^2)/(b))`
⇒ `sectheta = 1/(costheta) = b/(sqrt(b^2 - a^2))`
⇒ `tan theta = (sintheta)/(costheta) = a/(sqrt(b^2 - a^2))`
Now,
`sec theta + tan theta`
= `b/(sqrt(b^2 - a^2)) + a/(sqrt(b^2 - a^2))`
= `(b + a)/(sqrt((b - a)(b + a)))`
= `(sqrt(b + a)sqrt(b + a))/(sqrt(b - a)sqrt(b + a))`
= `(sqrt(b + a))/(sqrt(b - a))`
= `sqrt((b + a)/(b - a))`
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