Question

prove it please kindly.Its my humble request​

Answer 1

Answer:

Step-by-step explanation:

Looks Like there is some mistake in the question as the actual question is of the form to prove x²+y²+z²=r² from tan⁻¹ (yz/xr) +tan⁻¹ (zx/yr) +tan⁻¹ (xy/zr) =Π /2

It is equivalent to prove x²+y²+z²=r² from tan⁻¹ (yz/xr) +tan⁻¹ (zx/yr) +tan⁻¹ (xy/zr) =Π /2

tan⁻¹ (yz/xr) +tan⁻¹ (zx/yr) +tan⁻¹ (xy/zr) =Π /2 —-(1)

Assume,

tan⁻¹ (yz/xr)=A

tan⁻¹ (zx/yr)= B

tan⁻¹ (xy/zr) = C

Then eqn (1) becomes,

A+B+C= Π /2

A+B=(Π /2)-C

tan(A+B) =tan[(Π /2)-C]

[(tanA+tanB)]/[(1-tanA tanB)] = CotC

[(tanA+tanB)]/[(1-tanA tanB)] = 1/tanC —-(2)

But as per assumption,

tanA = (yz/xr)

tanB= (zx/yr)

tanC = (xy/zr)

Substituting in eqn (2)

[(yz/xr)+(zx/yr)]/[1-(yz/xr)(zx/yr)] = [1/(xy/zr)]

on simplification,

x²+y²=r²-z²

x²+y²+z²=r²