If x cos-y cos-2 -3-z cos-4 -3- prove that x y-y z-z x-0

Given xcosθ=y cos ( θ+2π/3 )=z cos ( θ+4π/3 )=k(say) x=k/cosθ y=k/cos [ θ+2π/3 ] z=k/cos [ θ+4π/3 ] xy+yz+zx=k^2 [ 1/cosθ cos ( θ+2π/3 ) +1/cos ( θ+2π/3 )cos ...

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Byju's Answer

Standard XII

Mathematics

Cosine Rule

If x cosθ=y c...

Question

If $x c o s θ = y c o s (θ + \frac{2 π}{3}) = z c o s (θ + \frac{4 π}{3})$ , prove that $x y + y z + z x = 0$

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Solution

Given $x c o s θ = y c o s (θ + \frac{2 π}{3}) = z c o s (θ + \frac{4 π}{3}) = k (s a y) x = \frac{k}{c o s θ} y = \frac{k}{c o s [θ + \frac{2 π}{3}]} z = \frac{k}{c o s [θ + \frac{4 π}{3}]}$

$x y + y z + z x = k^{2} [\frac{1}{c o s θ c o s (θ + \frac{2 π}{3})} + \frac{1}{c o s (θ + \frac{2 π}{3}) c o s (θ + \frac{4 π}{3})} + \frac{1}{c o s (θ + \frac{4 π}{3}) c o s θ}] = k^{2} [\frac{c o s (θ + \frac{4 π}{3}) + c o s θ + c o s (θ + \frac{2 π}{3})}{c o s θ c o s (θ + \frac{2 π}{3}) c o s (θ + \frac{4 π}{3})}] = k^{2} ⎡ ⎢ ⎣ \frac{c o s θ (\frac{- 1}{2}) - s i n θ (\frac{\sqrt{3}}{2}) + c o s θ + c o s θ (\frac{- 1}{2}) - s i n θ (\frac{\sqrt{3}}{2})}{c o s θ c o s (θ + \frac{2 π}{3}) c o s (θ + \frac{4 π}{3})} ⎤ ⎥ ⎦ = k^{2} ⎡ ⎢ ⎣ \frac{- c o s θ + s i n θ (\frac{\sqrt{3}}{2}) + c o s θ - s i n θ (\frac{\sqrt{3}}{2})}{c o s θ c o s (θ + \frac{2 π}{3}) c o s (θ + \frac{4 π}{3})} ⎤ ⎥ ⎦ = 0$

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Similar questions

Q. If

x cos θ = y cos (θ + \frac{2 π}{3}) = z cos (θ + \frac{4 π}{3})

, then

x y + y z + z x = 0.

State the whether given statement is true or false

x cos θ = y cos (θ + \frac{2 π}{3}) = z cos (θ + \frac{4 π}{3})

,then

x y + y z + z x = 0

Q. If

x \cos θ = y \cos (θ + \frac{2 π}{3}) = z \cos (θ + \frac{4 π}{3})

, prove that

x y + y z + z x = 0

. [NCERT EXEMPLAR]

Q. If

x, y, z

are non zero real numbers and if

x cos θ = y cos (θ + \frac{2 π}{3}) = z cos (θ + \frac{4 π}{3})

for some

θ \in R

, then show that

x y + y z + z x = 0

Q. If If

x c o s θ = y c o s (θ + \frac{2 π}{3}) = z c o s (θ + \frac{4 π}{3}),

then write the value of

\frac{1}{x} + \frac{1}{y} + \frac{1}{z}

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If xcosθ=y cos(θ+2π3)=z cos(θ+4π3), prove that xy+yz+zx=0

If $x c o s θ = y c o s (θ + \frac{2 π}{3}) = z c o s (θ + \frac{4 π}{3})$ , prove that $x y + y z + z x = 0$