Question

Sinteta/cosecteta+cot teta=2+sin teta/cot teta -cosecteta

Answer 1

$\textbf{To prove:}$

$\displaystyle\frac{sin\theta}{cosec\theta+cot\theta}=2+\frac{sin\theta}{cot\theta-cosec\theta}$

$\textbf{L.H.S}$

$=\displaystyle\frac{sin\theta}{cosec\theta+cot\theta}$

$\text{Multiply both numerator and denominator by cosec\theta-cot\theta }$

$=\displaystyle\frac{sin\theta}{cosec\theta+cot\theta}{\times}\frac{cosec\theta-cot\theta}{cosec\theta-cot\theta}$

$=\displaystyle\frac{sin\theta(cosec\theta-cot\theta)}{cosec^2\theta-cot^2\theta}$

$\text{Using, }\;\bf\;cosec^2A-cot^2A=1$

$=\displaystyle\frac{sin\theta(\frac{1}{sin\theta}-\frac{cos\theta}{sin\theta})}{1}$

$=1-cos\theta$

$\textbf{R.H.S}$

$=\displaystyle2+\frac{sin\theta}{cot\theta-cosec\theta}$

$=\displaystyle2-\frac{sin\theta}{cosec\theta-cot\theta}$

$\text{Multiply both numerator and denominator by cosec\theta+cot\theta }$

$=\displaystyle2-\frac{sin\theta}{cosec\theta-cot\theta}{\times}\frac{cosec\theta+cot\theta}{cosec\theta+cot\theta}$

$=\displaystyle2-\frac{sin\theta(cosec\theta+cot\theta)}{cosec^2\theta-cot^2\theta}$

$=\displaystyle2-\frac{sin\theta(\frac{1}{sin\theta}+\frac{cos\theta}{sin\theta})}{1}$

$=2-(1+cos\theta)$

$=1-cos\theta$

$\implies\textbf{L.H.S=R.H.S}$

$\text{Hence proved}$

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7 cosecA -3 cotA =7 prove that 7 cotA - 3 cosecA =3

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Answer 2

Answer:

\textbf{To prove:}To prove:

\displaystyle\frac{sin\theta}{cosec\theta+cot\theta}=2+\frac{sin\theta}{cot\theta-cosec\theta}cosecθ+cotθsinθ=2+cotθ−cosecθsinθ

\textbf{L.H.S}L.H.S

=\displaystyle\frac{sin\theta}{cosec\theta+cot\theta}=cosecθ+cotθsinθ

\text{Multiply both numerator and denominator by $cosec\theta-cot\theta$}Multiply both numerator and denominator by cosecθ−cotθ

=\displaystyle\frac{sin\theta}{cosec\theta+cot\theta}{\times}\frac{cosec\theta-cot\theta}{cosec\theta-cot\theta}=cosecθ+cotθsinθ×cosecθ−cotθcosecθ−cotθ

=\displaystyle\frac{sin\theta(cosec\theta-cot\theta)}{cosec^2\theta-cot^2\theta}=cosec2θ−cot2θsinθ(cosecθ−cotθ)

\text{Using, }\;\bf\;cosec^2A-cot^2A=1Using, cosec2A−cot2A=1

=\displaystyle\frac{sin\theta(\frac{1}{sin\theta}-\frac{cos\theta}{sin\theta})}{1}=1sinθ(sinθ1−sinθcosθ)

=1-cos\theta=1−cosθ

\textbf{R.H.S}R.H.S

=\displaystyle2+\frac{sin\theta}{cot\theta-cosec\theta}=2+cotθ−cosecθsinθ

=\displaystyle2-\frac{sin\theta}{cosec\theta-cot\theta}=2−cosecθ−cotθsinθ

\text{Multiply both numerator and denominator by $cosec\theta+cot\theta$}Multiply both numerator and denominator by cosecθ+cotθ

=\displaystyle2-\frac{sin\theta}{cosec\theta-cot\theta}{\times}\frac{cosec\theta+cot\theta}{cosec\theta+cot\theta}=2−cosecθ−cotθsinθ×cosecθ+cotθcosecθ+cotθ

=\displaystyle2-\frac{sin\theta(cosec\theta+cot\theta)}{cosec^2\theta-cot^2\theta}=2−cosec2θ−cot2θsinθ(cosecθ+cotθ)

=\displaystyle2-\frac{sin\theta(\frac{1}{sin\theta}+\frac{cos\theta}{sin\theta})}{1}=2−1sinθ(sinθ1+sinθcosθ)

=2-(1+cos\theta)=2−(1+cosθ)

=1-cos\theta=1−cosθ

\implies\textbf{L.H.S=R.H.S}⟹L.H.S=R.H.S

\text{Hence proved}Hence proved

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7 cosecA -3 cotA =7 prove that 7 cotA - 3 cosecA =3

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