## PrecalcIsNotMyFriend one year ago Identities- Ugh! 1÷cscθ−cotθ=1+cosθ÷sinθ

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1. Tushara

r u missing any brackets?

2. ZeHanz

It should be:$\frac{ 1 }{ \csc \theta - \cot \theta }=\frac{ 1+\cos \theta }{ \sin \theta }$Let's begin at the left hand side:$\frac{ 1 }{ \csc \theta - \cot \theta }=\frac{ 1 }{ \frac{ 1 }{ \sin \theta }-\frac{ \cos \theta }{ \sin \theta } }=\frac{ 1 }{ \frac{ 1-\cos \theta }{ \sin \theta } }=\frac{ \sin \theta }{ 1- \cos \theta }$This looks much better, although it is not quite the right hand side yet... Hint: multiply numerator and denominator of the last fraction with (1+cosθ)/(1+cosθ):$\frac{ \sin θ }{ 1-\cos θ } \cdot \frac{ 1+\cos θ }{ 1+\cos θ }= ...$After just a few small steps, you'll have got to the right hand side of the identity!