Question

(cosec theta + 1/ cot theta)² = cosec theta+ 1/ cosec theta- 1

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Answer 1

$\underline{\underline{\maltese\: \: \textbf{\textsf{Question}}}}$

$\sf ({{cosec \theta - }{cot \theta})}² = \frac{1 - cos \theta}{1 + cos \theta}$

$\underline{\underline{\maltese\: \: \textbf{\textsf{Answer}}}}$

LHS

$\sf ({{cosec \theta - }{cot \theta})}²$

$\begin{gathered} \sf \longrightarrow(\frac{1}{sin\theta}-\frac{cos\theta}{sin\theta}\big)^{2}\end{gathered}$

$\longrightarrow \sf \large ({\frac{ {1 - cos \theta} }{sin \theta}})^{2}$

$\longrightarrow \sf \large\frac{ ({1 - cos \theta})^{2} }{ ({sin \theta})^{2} }$

$\longrightarrow \sf \large \frac{ ({1 - cos \theta})^{2} }{1 - {cos}^{2} \theta }$

$\longrightarrow \sf \large\frac{(1-cos\theta)(1-cos\theta)}{(1+cos\theta)(1-cos\theta)}$

$\begin{gathered} \longrightarrow \sf \frac{1-cos\theta}{1+cos\theta}\end{gathered}$

=RHS

Additional Information :-

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

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

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

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

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

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

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

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

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