If n sin θ = m cos θ , then prove that
Answers
Answer:
\huge{\red{\boxed{ \overline{\underline{ \mid \tt{ \red{Your \: Question. \red { \mid}}}}}}}}
∣YourQuestion.∣
\rm \: If \: n \: sin \: \theta = m \: cos \: \theta \: then, \: prove \: that \:Ifnsinθ=mcosθthen,provethat
\begin{lgathered}\sf{\frac{m \: sin \: \theta - n \: cos \: \theta}{m \: sin \: \theta + \: n \: cos \: \theta} + \frac{m \: sin \: \theta + n \: cos \: \theta}{m \: sin \: \theta \: - n \: cos \: \theta}} \\\end{lgathered}
msinθ+ncosθ
msinθ−ncosθ
+
msinθ−ncosθ
msinθ+ncosθ
\begin{lgathered}\sf = \frac{2(m {}^{4} + n {}^{4})}{m {}^{4} - n {}^{4} }. \\\end{lgathered}
=
m
4
−n
4
2(m
4
+n
4
)
.
\huge{\red{\boxed{\overline{\underline{ \mid \tt{ \red{Solution. \red{ \mid}}}}}}}}
∣Solution.∣
\begin{lgathered}\bf \blue{G} \red{I} \purple{V} \green{E} \orange{N} \\\end{lgathered}
GIVEN
\rm \: n \: sin \: \theta = m \: cos \: \thetansinθ=mcosθ
\begin{lgathered}\implies \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \rm \: sin \: \theta = \frac{m}{n} \: cos \: \theta \\\end{lgathered}
⟹sinθ=
n
m
cosθ
\: \: \: \rm \: Putting \: this \: value \: of \: sin \: \theta \: in \: LHS,PuttingthisvalueofsinθinLHS,
\begin{lgathered}\therefore \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \rm \: LHS = \frac{m \: sin \: \theta - n \: cos \ \theta}{m \: sin \: \theta + n \: cos \: \theta} + \frac{ m \: sin \: \theta + n \: cos \: \theta}{m \: sin \: \theta \: - \: n \: cos \: \theta} \\\end{lgathered}
∴LHS=
msinθ+ncosθ
msinθ−ncos θ
+
msinθ−ncosθ
msinθ+ncosθ
\begin{lgathered}\rm = \frac{m \bigg( \frac{m}{n} \: cos \: \theta \bigg) - \: n \: cos \: \theta} {m \bigg( \frac{m}{n} \: cos \: \theta \bigg) + \: n \: cos \: \theta} + \frac{m \bigg( \frac{m}{n} \: \: cos \: \theta \bigg) \: - \: n \: cos \: \theta }{m \bigg( \frac{m}{n} \: cos \: \theta \bigg) - \: n \: cos \: \theta} \\\end{lgathered}
=
m(
n
m
cosθ)+ncosθ
m(
n
m
cosθ)−ncosθ
+
m(
n
m
cosθ)−ncosθ
m(
n
m
cosθ)−ncosθ
\begin{lgathered}= \: \rm \frac {cos \ \theta \bigg( \frac { m {}^{2} } {n} - n \bigg) }{cos \: \theta \bigg (\frac{m {}^{2} }{n } + {n \bigg)} } + \ \: \frac { cos \ \theta \bigg(\frac{m {}^{2} } n {}^{} + \: n \bigg) }{cos \ \theta \bigg(\frac{ m{}^{2} }{n } - {n \bigg)} } \\\end{lgathered}
=
cosθ(
n
m
2
+n)
cos θ(
n
m
2
−n)
+
cos θ(
n
m
2
−n)
cos θ(
n
m
2
+n)
\begin{lgathered}= \: \: \: \rm \frac{ \frac{m {}^{2} } {n} - n }{\frac{m {}^{2} }{n } + {n} } + \: \frac{ \frac{m {}^{2} } n {}^{} + \: n }{\frac{m{}^{2} }{n } - {n} } \\\end{lgathered}
=
n
m
2
+n
n
m
2
−n
+
n
m
2
−n
n
m
2
+n
\begin{lgathered}\rm = \: \: \: \: \: \: \: \: \frac{ \frac{m {}^{2} - n {}^{2} } {n} }{\frac{m {}^{2} + n {}^{2} }{n}} + \frac{ \frac{m {}^{2} + n {}^{2} }{n} } {\frac{m {}^{2} - n {}^{2} }{n}} \\\end{lgathered}
=
n
m
2
+n
2
n
m
2
−n
2
+
n
m
2
−n
2
n
m
2
+n
2
\begin{lgathered}= \: \: \: \: \: \rm \frac{m {}^{2} - n {}^{2} }{m {}^{2} + n {}^{2} } + \frac{m {}^{2} + n {}^{2} }{m {}^{2} - n {}^{2} } \\\end{lgathered}
=
m
2
+n
2
m
2
−n
2
+
m
2
−n
2
m
2
+n
2
\: \: \: \: \: \: \: \: \: \rm \: Taking \: L.C.M.,TakingL.C.M.,
\begin{lgathered}\rm = \frac{(m {}^{2} - n {}^{2} ) {}^{2} + (m {}^{2} + n {}^{2} ) {}^{2}}{(m {}^{2} + n {}^{2})(m {}^{2} - n {}^{2})} \\\end{lgathered}
=
(m
2
+n
2
)(m
2
−n
2 )
(m
2
−n
2
)
2
+(m
2
+n
2
)
2
\begin{lgathered}\rm = \: \frac{(m {}^{4} + n {}^{4} - 2m {}^{2} n {}^{2} ) + (m {}^{4} + n {}^{4} + 2m {}^{2} n {}^{2})}{m {}^{4} - n {}^{4}} \\\end{lgathered}
=
m
4
−n
4
(m
4
+n
4
−2m
2
n
2
)+(m
4
+n
4
+2m
2
n
2
)
\begin{lgathered}\rm = \: \: \frac{ 2(m {}^{4} + n {}^{4} )}{m {}^{4} - n {}^{4}} = RHS \\\end{lgathered}
=
m
4
−n
4
2(m
4
+n
4
)
=RHS