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 } } = \, ?

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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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Angela Fajardo by Angela Fajardo , 19, Philippines

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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

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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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