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Trigonometric Identities

If x/acos θ+y...

Question

$If \frac{x}{a} c o s θ + \frac{y}{b} s i n θ = 1 and \frac{x}{a} s i n θ - \frac{y}{b} c o s θ = 1, prove that \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 2$

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Solution

---------- (i)
---------- (ii)
Square both equations and add
$equals x squared over a squared cos squared theta plus y squared over b squared sin squared theta space plus 2 space x over a cos theta space y over b sin theta plus space fraction numerator begin display style x squared end style over denominator begin display style a squared end style end fraction sin squared theta plus fraction numerator begin display style y squared end style over denominator begin display style b squared end style end fraction cos squared theta minus 2 x over a sin theta fraction numerator y over denominator b space end fraction cos theta equals x squared over a squared left parenthesis sin squared theta plus cos squared theta right parenthesis plus y squared over b squared left parenthesis sin squared theta plus cos squared theta right parenthesis space equals fraction numerator begin display style x squared end style over denominator begin display style a squared end style end fraction plus fraction numerator begin display style y squared end style over denominator begin display style b squared end style end fraction$
LHS = RHS
Hence proved

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Similar questions

x = a sin θ + b cos θ

y = a cos θ - b sin θ

, then prove that

x^{2} + y^{2} = a^{2} + b^{2}

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y = a sin θ - b cos θ

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t a n θ + s i n θ = m

t a n θ - s i n θ = n

m^{2} - n^{2} = 4 \sqrt{m n}

\frac{x}{a} s i n θ + \frac{y}{b} c o s θ = 1

\frac{x}{a} c o s θ - \frac{y}{b} s i n θ = 1

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 2

If $a cos θ - b sin θ = c$ , prove that $a sin θ + b cos θ = \pm \sqrt{a^{2} + b^{2} - c^{2}}$

Q. If a cos θ + b sin θ = m and a sin θ − b cos θ = n, prove that (m² + n²) = (a² + b²).

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