cos theta is equal to 2 cot theta + sin theta upon 1 + cos theta is equal to 2 show that
Answers
Step-by-step explanation:
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Answer:
\begin{gathered}If \: cosec\theta = 2 \: then\\ cot\theta+\frac{sin\theta}{(1+cos\theta)}=2\end{gathered}
Ifcosecθ=2then
cotθ+
(1+cosθ)
sinθ
=2
Step-by-step explanation:
Given \: cosec\theta = 2 --(1)Givencosecθ=2−−(1)
LHS = cot\theta+\frac{sin\theta}{(1+cos\theta)}LHS=cotθ+
(1+cosθ)
sinθ
=cot\theta+\frac{sin\theta(1-cos\theta)}{(1+cos\theta)(1-cos\theta)}=cotθ+
(1+cosθ)(1−cosθ)
sinθ(1−cosθ)
=cot\theta+\frac{sin\theta(1-cos\theta)}{1^{2}-cos^{2}\theta}=cotθ+
1
2
−cos
2
θ
sinθ(1−cosθ)
=cot\theta+\frac{sin\theta(1-cos\theta)}{sin^{2}\theta}=cotθ+
sin
2
θ
sinθ(1−cosθ)
=cot\theta+\frac{(1-cos\theta)}{sin\theta}=cotθ+
sinθ
(1−cosθ)
=cot\theta+\frac{1}{sin\theta}-\frac{cos\theta}{sin\theta}=cotθ+
sinθ
1
−
sinθ
cosθ
=cot\theta+cosec\theta-cot\theta=cotθ+cosecθ−cotθ
\begin{gathered}=cosec\theta\\=2\:[From\:(1)]\\=RHS\end{gathered}
=cosecθ
=2[From(1)]
=RHS
Therefore,
\begin{gathered}If \: cosec\theta = 2 \: then\\ cot\theta+\frac{sin\theta}{(1+cos\theta)}=2\end{gathered}
Ifcosecθ=2then
cotθ+
(1+cosθ)
sinθ
=2
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