If x- a sin- - c cos- and y- a cos- - c sin- then x2-y2- a2- c2-

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Byju's Answer

Standard XII

Mathematics

Basic Trigonometric Identities

If x= a sin...

Question

If $x = a sin θ + c cos θ$ and $y = a cos θ - c sin θ$ , then $x^{2} + y^{2} = a^{2} + c^{2}$ .

True

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False

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Solution

The correct option is A True
$x^{2} + y^{2} = (a s i n θ + c c o s θ)^{2} + (a c o s θ - c s i n θ)^{2} = a^{2} s i n^{2} θ + c^{2} c o s^{2} θ + 2 a c s i n θ c o s θ + a^{2} c o s^{2} θ + c^{2} s i n^{2} θ - 2 a c s i n θ c o s θ = a^{2} (s i n^{2} θ + c o s^{2} θ) + c^{2} (s i n^{2} θ + c o s^{2} θ) = a^{2} + c^{2}$

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If x=asinθ+ccosθ and y=acosθ−csinθ, then x2+y2=a2+c2.

The correct option is A Truex2+y2=(asinθ+ccosθ)2+(acosθ−csinθ)2=a2sin2θ+c2cos2θ+2acsinθcosθ+a2cos2θ+c2sin2θ−2acsinθcosθ=a2(sin2θ+cos2θ)+c2(sin2θ+cos2θ)=a2+c2

If $x = a sin θ + c cos θ$ and $y = a cos θ - c sin θ$ , then $x^{2} + y^{2} = a^{2} + c^{2}$ .