Question

d/dx (sin x + cos x / sin x - cos x )

Answer 1

Answer:

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Answer 2

${ \bf{ \underline{ \red{ \underline{ \red{solution}}}} : - }}$

Let,

${\dashrightarrow { \bf{ \purple { y \: = \frac{sin \: x \: + \: cos \: x}{sin \: x \: - \: cos \: x} }}}}$

Rationlize denominator { by taking conjugate pair of denominator }

${\dashrightarrow { \bf{ \purple { y \: = \frac{sin \: x \: + \: cos \: x}{sin \: x \: - \: cos \: x} \times \frac{sin \: x \: + \: cos \: x}{sin \: x \: + \: cos \: x} }}}}$

${\dashrightarrow { \bf{ \purple { y \: = \frac{(sin \: x \: + \: cos \: x) \times (sin \: x \: + \: cos \: x) }{(sin \: x \: - \: cos \: x) \times (sin \: x \: + \: cos \: x) } }}}}$

${\dashrightarrow { \bf{ \purple { y \: = \frac{ {sin}^{2}x +{ {sin \: x \times cos \: x + sin \: x \times cos \: x }}+ {cos}^{2}x }{ {sin}^{2}x +{ { \cancel{sin \: x \times cos \: x }- { \cancel{sin \: x \times cos \: x }}} }- {cos}^{2}x } }}}}$

${\dashrightarrow { \bf{ \purple { y \: = \frac{ {sin}^{2}x +2(sin \: x \times cos \: x ) + {cos}^{2}x }{ {sin}^{2}x - {cos}^{2}x } }}}}$

${\dashrightarrow { \bf{ \purple { y \: = \frac{ {sin}^{2}x + {cos}^{2}x + 2(sin \: x \times cos \: x ) }{ {sin}^{2}x - {cos}^{2}x } }}}}$

Formula: sin²x + cos²x = 1

${\dashrightarrow { \bf{ \purple { y \: = \frac{1 + 2(sin \: x \times cos \: x ) }{ {sin}^{2}x - {cos}^{2}x } }}}}$

Formula : 2 ( sin x • cos x ) = sin 2x

${\dashrightarrow { \bf{ \purple { y \: = \frac{ 1 \: + \: sin \: 2x}{ {sin}^{2} x - {cos}^{2} x} }}}}$

${\dashrightarrow { \bf{ \purple { y \: = \frac{ 1 \: + \: sin \: 2x}{ - ({cos}^{2}x - {sin}^{2} x)} }}}}$

Formula : cos²x - sin²x = cos 2x

${\dashrightarrow { \bf{ \purple { y \: = \frac{ 1 \: + \: sin \: 2x}{ - \: cos \: 2x} }}}}</p><p>$

${\dashrightarrow { \bf{ \purple { y \: = { - \frac{1}{cos \: 2x} } + \frac{sin \:2 x}{cos \: 2x} }}}}$

${\dashrightarrow { \bf{ \purple { y \: = { - \: sec \: 2x \: + \: tan \: 2x}}}}}</p><p>$

Now,

${\dashrightarrow { \bf{ \purple { \frac{dy }{dx} \: = { - \: sec \: 2x \: + \: tan \: 2x}}}}}</p><p>$

${\dashrightarrow { \bf{ \purple { \frac{dy }{dx} \: = { - \: 2 \: sec \:2 x \times \: tan \: 2x + 2 \: {sec}^{2} \: 2 x}}}}}</p><p>$

${\dashrightarrow { \bf{ \purple { \frac{dy }{dx} \: = { - \: 2 \: sec \:2x ( \: tan \:2x + {sec} \: 2 x)}}}}}</p><p>$