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प्रश्न
If x = r sin A cos B, y = r sin A sin B and z = r cos A, then prove that : x2 + y2 + z2 = r2
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उत्तर
L.H.S. = x2 + y2 + z2
= (r sin A cos B)2 + (r sin A sin B)2 + (r cos A)2
= r2 sin2 A cos2 B + r2 sin2 A sin2 B + r2 cos2 A
= r2 sin2 A (cos2 B + sin2 B) + r2 cos2 A
= r2 (sin2 A + cos2 A)
= r2 = R.H.S.
संबंधित प्रश्न
Prove the following trigonometric identities:
`(1 - cos^2 A) cosec^2 A = 1`
Prove the following identities:
cot2 A – cos2 A = cos2 A . cot2 A
Prove the following identities:
`sqrt((1 - cosA)/(1 + cosA)) = cosec A - cot A`
Prove the following identities:
`1 - sin^2A/(1 + cosA) = cosA`
Prove that:
`(sinA - cosA)(1 + tanA + cotA) = secA/(cosec^2A) - (cosecA)/(sec^2A)`
Write the value of `sin theta cos ( 90° - theta )+ cos theta sin ( 90° - theta )`.
If a cos θ – b sin θ = c, then prove that (a sin θ + b cos θ) = `± sqrt(a^2 + b^2 - c^2)`
Choose the correct alternative:
Which is not correct formula?
Prove that `(sin^2theta)/(cos theta) + cos theta` = sec θ
Show that, cotθ + tanθ = cosecθ × secθ
Solution :
L.H.S. = cotθ + tanθ
= `cosθ/sinθ + sinθ/cosθ`
= `(square + square)/(sinθ xx cosθ)`
= `1/(sinθ xx cosθ)` ............... `square`
= `1/sinθ xx 1/square`
= cosecθ × secθ
L.H.S. = R.H.S
∴ cotθ + tanθ = cosecθ × secθ
