Question

prove that cos 37°+ sin37
cos 370-sin 37°
cot 8º.​

Answer 1

$\Huge \underline{\mathcal \pink{A}\purple{\frak{nSwer}}}$

$\sf{\begin{gathered} LHS = \frac{cos 37\degree + sin 37 \degree }{cos 37\degree - sin 37 \degree} \\= \frac{cos 37\degree + sin (90 - 53) \degree }{cos 37\degree - sin (90- 53) \degree} \\= \frac{cos 37\degree + cos 53 \degree }{cos 37\degree - cos 53 \degree} \\= \frac{ 2 cos \Big( \frac{37+53}{2}\Big) cos \Big( \frac{37-53}{2}\Big) }{ -2 sin \Big( \frac{37+53}{2}\Big) sin \Big( \frac{37-53}{2}\Big) }\\= \frac{ 2 cos \Big( \frac{90}{2}\Big) cos \Big( \frac{-16}{2}\Big) }{ -2 sin \Big( \frac{90}{2}\Big) sin \Big( \frac{-16}{2}\Big) }\\= \frac{ cos 45 \degree cos (-8) \degree }{ - sin 45 \degree sin (-8) \degree } \\= \frac{\frac{1}{\sqrt{2}} \times cos 8 \degree }{ - \frac{1}{\sqrt{2}} \times ( - sin \: 8 \degree ) }\\= \frac{cos 8 \degree }{ sin 8 \degree } \\= cot 8 \degree \\ = RHS \end{gathered}}$

Hence proved

Answer 2

$\large\underline{\bold{Given \:Question - }}$

$\: \: \: \: \: \: \: \: \bull \: \sf \: \: \: Prove \: that \: \dfrac{cos37 \degree \: + \: sin37\degree}{cos37\degree - sin37\degree} = cot8\degree$

$\large\underline{\sf{Solution-}}$

We know that

$\boxed{ \bf \: cot(x - y) = \dfrac{cotx \: coty \: + \: 1}{coty - cotx}}$

Now,

Consider LHS,

$\rm :\longmapsto\:\dfrac{cos37\degree + sin37\degree}{cos37\degree - sin37\degree}$

Divide numerator and denominator by sin37°, we get

$\sf \: \: \: = \: \: \: \dfrac{\dfrac{cos37\degree}{sin37\degree} + \dfrac{sin37\degree}{sin37\degree} }{ \: \: \: \: \dfrac{cos37\degree}{sin37\degree} - \dfrac{sin37\degree}{sin37\degree} \: \: \: \: }$

$\sf \: \: \: = \: \: \: \dfrac{cot37\degree + 1}{cot37\degree - 1}$

$\sf \: \: \: = \: \: \: \dfrac{cot37\degree \times 1 + 1}{cot37\degree - 1}$

$\sf \: \: \: = \: \: \: \dfrac{cot37\degree \times cot45\degree + 1}{cot37\degree - cot45\degree} \: \: \: \{ \because \: cot45\degree = 1 \}$

$\sf \: \: \: = \: \: \: cot(45\degree - 37\degree)$

$\sf \: \: \: = \: \: \: cot8\degree$

${\boxed{\boxed{\bf{Hence, Proved}}}}$

Aliter Method

Consider RHS,

$\rm :\longmapsto\:cot8\degree$

$\sf \: \: \: = \: \: \: cot(45\degree - 8\degree)$

$\sf \: \: \: = \: \: \: \dfrac{cot45\degree \: cot37\degree \: + 1}{cot37\degree - cot45\degree}$

$\sf \: \: \: = \: \: \: \dfrac{1 \times \: cot37\degree \: + 1}{cot37\degree - 1}$

$\sf \: \: \: = \: \: \: \dfrac{ \: cot37\degree \: + 1}{ \: \: \: \: cot37\degree - 1 \: \: \: \: }$

$\sf \: \: \: = \: \: \: \dfrac{\dfrac{cos37\degree}{sin37\degree} + 1}{\dfrac{cos37\degree}{sin37\degree} - 1 }$

$\sf \: \: \: = \: \: \: \dfrac{\dfrac{cos37\degree + sin37\degree}{ \cancel{sin37\degree}}}{ \: \: \: \: \dfrac{cos37\degree - sin37\degree}{ \cancel{sin37\degree}} \: \: \: \: }$

$\sf \: \: \: = \: \: \: \dfrac{cos37\degree + sin37\degree}{cos37\degree - sin37\degree}$

${\boxed{\boxed{\bf{Hence, Proved}}}}$

Additional Information :-

$\boxed{ \sf \: sin(x + y) = sinxcosy + sinycosx}$

$\boxed{ \sf \: sin(x - y) = sinxcosy - sinycosx}$

$\boxed{ \sf \: cos(x + y) = cosxcosy - sinxsiny}$

$\boxed{ \sf \: cos(x - y) = cosxcosy + sinxsiny}$

$\boxed{ \sf \: tan(x + y) = \dfrac{tanx + tany}{1 - tanx \: tany}}$

$\boxed{ \sf \: tan(x - y) = \dfrac{tanx - tany}{1 + tanx \: tany}}$

$\boxed{ \sf \: cot(x + y) = \dfrac{cotxcoty - 1}{coty + cotx}}$

$\boxed{ \sf \: {sin}^{2} x - {sin}^{2} y = sin(x + y)sin(x - y)}$

$\boxed{ \sf \: {cos}^{2} x - {sin}^{2} y = cos(x + y) \: cos(x - y)}$