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If x=asecthetacosvarphi,\ \ y=bsecthetas...

If `x=asecthetacosvarphi,\ \ y=bsecthetasinvarphi` and `z=ctantheta` , show that `(x^2)/(a^2)+(y^2)/(b^2)-(z^2)/(c^2)=1`

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We have
` (x)/(a) = sec theta cos phi " "...(i) and (y)/(b) = sec theta sin phi " "...(ii). `
Squaring (i) and (ii) and adding, we get
` ((x^(2))/(a^(2)) + (y^(2))/(b^(2))) = sec^(2) theta (cos^(2) phi + sin^(2) phi )`
` = sec^(2) theta " "[ because cos^(2) phi + sin^(2) phi = 1] `
` = (1+ tan^(2) theta ) = (1+ (z^(2))/(c^(2))) " " [ because tan theta = (z)/(c) ] `
Hence, `((x^(2))/(a^(2)) + (y^(2))/(b^(2)) ) = (1+ (z^(2))/(c^(2))). `
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RS AGGARWAL-TRIGONOMETRIC IDENTITIES-Test Yourself
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  5. If sin theta=(sqrt(3))/(2)" then " ("cosec"theta+cot theta)=?

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  6. If cotA=(4)/(5), then ((sinA+cosA))/((sinA-cosA))=?

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  7. If 2x=secA and (2)/(x)=tanA, " prove that " (x^(2)-(1)/(x^(2)))=(1)/(4...

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  11. Prove that sqrt((1+cosA)/(1-cosA))=("cosec"A+cotA).

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  12. If cosec theta + cot theta= p, cos theta = ?

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  13. Prove that ("cosec"A-cotA)^(2)=((1-cosA))/((1+cosA)).

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  14. If 5cottheta=3. find the value of (5sintheta-3costheta)/(4sintheta+3co...

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  15. Prove that (sin32^(@)cos58^(@)+cos32^(@)sin58^(@))=1.

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  16. If x=a sin theta+b cos theta and y=a cos theta -b sin theta, " prove t...

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