Question

If x=a sin theta + bcos theta and y= a cos theta + b sin theta prove that x² + y² = a ² + b²

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Answer 1

⠀⠀$\huge\underline{ \overline{ \bf{ \purple{QUESTION}}}}$

correct question=

If x=a sin theta + bcos theta and y= a cos theta - b sin theta prove that x² + y² = a ² + b²

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⠀⠀⠀⠀⠀$\huge\underline{ \underline{ \bf{ { \blue{ an{ \pink{sw{ \purple{er \: : = }}}}} }}}}$

$\large\underline{ \underline{ \green{ \bold {given}}}} = >$

⠀⠀⠀⠀${ \bf{ \red{x = a \: sin \theta + b \: cos\theta}}} \\ { \bf{ \red{y = a \: cos\theta + b \: sin \theta \: }}}$

$\large\underline{ \underline{ \green{ \bold{solution}}}} = >$

Now,

We have,

x² + y²

⟹ ( a cos ∅ + b sin ∅ )² + ( a sin∅ - b cos∅ )² .⠀⠀

⟹a²cos²∅ + b²cos²∅ + 2 ab cos∅ sin∅ + a²sin²∅ + b²sin²∅ - 2 ab sin∅ cos∅ .

⟹a²cos²∅ + b²cos²∅ + a²sin²∅ + b²sin²∅ .⠀

⟹a²cos²∅ + a²sin²∅ + b²cos²∅ + b²cos²∅ .⠀⠀

➠ a²( cos²∅ + sin²∅ ) + b²( cos²∅ + sin²∅ ) .

⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀= a² + b²

⠀So,

⠀⠀⠀⠀⠀⠀ x² + y² =a² + b²

⠀⠀⠀⠀⠀⠀⠀⠀LHS = RHS .

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Answer 2

⠀⠀$\huge\underline{ \mathbb{ \pink{QUESTION}}}$

correct question=

If x=a sin theta + bcos theta and y= a cos theta - b sin theta prove that x² + y² = a ² + b²

_________________________________

⠀⠀⠀⠀⠀$\huge {\mathbf{ \green{ \underbrace{ \blue{ \boxed{ \red{ \fbox{ \green{ \underline{ answer}}}}}}}}}}$

$\large\underline{ \underline{ \blue{ \bold {given}}}} = >$

${ \bf{ \pink{x = a \: sin \theta + b \: cos\theta}}} \\ { \bf{ \red{y = a \: cos\theta + b \: sin \theta \: }}}$

$\large\underline{ \underline{ \blue{ \bold{solution}}}} = >$

Now,

We have,

x² + y²

⟹ ( a cos ∅ + b sin ∅ )² + ( a sin∅ - b cos∅ )² .⠀⠀

⟹a²cos²∅ + b²cos²∅ + 2 ab cos∅ sin∅ + a²sin²∅ + b²sin²∅ - 2 ab sin∅ cos∅ .

⟹a²cos²∅ + b²cos²∅ + a²sin²∅ + b²sin²∅ .⠀

⟹a²cos²∅ + a²sin²∅ + b²cos²∅ + b²cos²∅ .⠀⠀

➠ a²( cos²∅ + sin²∅ ) + b²( cos²∅ + sin²∅ ) .

⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀= a² + b²

⠀So,⠀⠀⠀⠀

⠀⠀⠀⠀ x² + y² =a² + b²

LHS = RHS .

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