Question

I will mark BRAINLIEST for the first correct answer. Plz solve it fast..........

Answer 1

Answer

$\sf \dfrac{ {x}^{2} + {y}^{2} }{ {x}^{2} - {y}^{2} }$

Given

• $\sf{\tan \theta = \frac{x}{y} }$

To Find

The value of

• $\sf \dfrac{x \sin \theta + y \cos \theta}{x \sin \theta - y \cos \theta}$

Solution

Taking the given value

$\sf \tan \theta = \dfrac{x}{y}$

$\implies \sf\dfrac{ \sin \theta}{ \cos \theta} = \dfrac{x}{y}$

$\implies \sf\sin \theta = \dfrac{x}{y} \cos \theta$

Now using the value of sinθ in the expression which is given to find :

$\sf \longrightarrow \dfrac{x \times \frac{x}{y}\cos \theta + y \cos \theta}{x \times \frac{x}{y} \cos \theta - y\cos \theta}$

$\sf\longrightarrow \dfrac{ \dfrac{ {x}^{2} \cos \theta + {y}^{2} \cos \theta }{ y} }{ \dfrac{{x}^{2} \cos \theta - {y}^{2} \cos \theta}{y} }$

$\longrightarrow \sf \dfrac{ \cos \theta( {x}^{2} + {y}^{2} )}{ \cos \theta( {x}^{2} - {y}^{2}) }$

$\sf \longrightarrow \dfrac{ {x}^{2} + {y}^{2} }{ {x}^{2} - y {}^{2} }$

Answer 2

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QUESTION :

I will mark BRAINLIEST for the first correct answer. Plz solve it fast..........

$\tan( \theta ) = \dfrac{X}{Y} then\; find \:\dfrac{x \sin ( \theta ) + y \cos ( \theta ) }{x \sin ( \theta ) - y \cos ( \theta ) }$

CONCEPT USED :

Ratio And Proportion , Trigonometric Identities

SOLUTION :

$\tan( \theta ) = \dfrac{X}{Y} \\ \\ \dfrac{ \sin ( \theta ) }{ \cos (\theta ) } = \dfrac{X}{Y}$

So :

$Suppose \: \sin ( \theta ) = X \: And \: \cos ( \theta ) = Y$

Note that the above sentence can be written by ratio and proportion.

So, $\dfrac{x \sin ( \theta ) + y \cos ( \theta ) }{x \sin ( \theta ) - y \cos ( \theta ) }$

$=> \dfrac { { x } ^ 2 + { y } ^ 2}{{ x } ^ 2 - { y } ^ 2}........ ( A )$