Break Me Into Sines And Cosines

Geometry Level 1

csc θ sin θ cot θ tan θ = ? \large \dfrac { \csc { \theta } }{ \sin { \theta } } - \dfrac { \cot { \theta } }{ \tan { \theta } } = \, ?


Check out the set: 2016 Problems

sin θ cos θ \sin { \theta } \cos { \theta } cos θ cot θ \cos { \theta } \cot { \theta } tan θ \tan { \theta } 1 1

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2 solutions

It simplifies to -

c o s e c 2 θ c o t 2 θ cosec^2\theta - cot^2\theta = 1

I remember my high school days heheheh!!

csc ( θ ) sin ( θ ) cot ( θ ) tan ( θ ) = 1 sin ( θ ) sin ( θ ) 1 tan ( θ ) tan ( θ ) = 1 sin 2 ( θ ) 1 tan 2 ( θ ) = 1 sin 2 ( θ ) cos 2 ( θ ) sin 2 ( θ ) = 1 cos 2 ( θ ) sin 2 ( θ ) = sin 2 ( θ ) sin 2 ( θ ) = 1 \displaystyle{\begin{aligned} \frac{\csc(\theta)}{\sin(\theta)} - \frac{\cot(\theta)}{\tan(\theta)} &= \frac{\frac{1}{\sin(\theta)}}{\sin(\theta)} - \frac{\frac{1}{\tan(\theta)}}{\tan(\theta)} \\&= \frac{1}{\sin^2(\theta)} - \frac{1}{\tan^2(\theta)} \\&= \frac{1}{\sin^2(\theta)} - \frac{\cos^2(\theta)}{\sin^2(\theta)} \\&= \frac{1-\cos^2(\theta)}{\sin^2(\theta)} \\&= \frac{\sin^2(\theta)}{\sin^2(\theta)} \\& = 1 \space\space\space\space\space \square \end{aligned}}

There are many ways to solve this given. to make it simpler.

FIN!!!!! \large \text{FIN!!!!!}

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