If x = r cos A cos B, y = r cos A sin B and z = r sin A , show that : x 2 + y 2 + z 2 = r 2

To prove: x 2 + y 2 + z 2 = r 2 Substituting value of x, y and z in L.H.S. of the equation : = (r cos A cos B) 2 + (r cos A sin B) 2 + (r sin A) 2 = r 2 cos 2 A cos 2 B + r 2 cos 2 A sin 2 B + r 2 sin 2 A = r 2 cos 2 A(cos 2 B + sin 2 B) + r 2 sin 2 A = r 2 cos 2 A + r 2 sin 2 A [∵ sin 2 θ + cos 2 θ = 1] = r 2 (cos 2 A + sin 2 A) = r 2 × 1 [∵ sin 2 θ + cos 2 θ = 1] = r 2 . Since, L.H.S. = R.H.S. Hence, proved that x 2 + y 2 + z 2 = r 2 .

Mathematics

If x = r cos A cos B, y = r cos A sin B and z = r sin A , show that :
x² + y² + z² = r²

Trigonometric Identities

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Answer

To prove:

x² + y² + z² = r²

Substituting value of x, y and z in L.H.S. of the equation :

= (r cos A cos B)² + (r cos A sin B)² + (r sin A)²

= r² cos² A cos² B + r² cos² A sin² B + r² sin² A

= r²cos² A(cos² B + sin² B) + r² sin² A

= r²cos² A + r²sin² A [∵ sin² θ + cos² θ = 1]

= r²(cos² A + sin² A)

= r² × 1 [∵ sin² θ + cos² θ = 1]

= r².

Since, L.H.S. = R.H.S.

Hence, proved that x² + y² + z² = r².

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Mathematics

If x = r cos A cos B, y = r cos A sin B and z = r sin A , show that :
x² + y² + z² = r²

Trigonometric Identities

Answer

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