It's trigo time #2

Geometry Level 2

Which of the following options is equal to:

csc ⁡ 2 θ − 1 csc ⁡ 2 θ \large \frac{\csc^2\theta-1}{\csc^2\theta}

Bonus: Prove it.

This is a part of a set: It's trigo time

sin ⁡ 2 θ \sin^2\theta -1 csc ⁡ 2 θ \csc^2\theta 1 0 cos ⁡ 2 θ \cos^2\theta

Syed Hamza Khalid by Syed Hamza Khalid , 17, France

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2 solutions

csc ⁡ 2 θ − 1 csc ⁡ 2 θ = csc ⁡ 2 θ csc ⁡ 2 θ − 1 csc ⁡ 2 θ = 1 − sin ⁡ 2 θ = cos ⁡ 2 θ \large\dfrac{\csc^2 \theta - 1}{\csc^2 \theta}=\dfrac{\csc^2 \theta}{\csc^2 \theta}-\dfrac{1}{\csc^2 \theta}=1-\sin^2 \theta=\cos^2 \theta

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Chew-Seong Cheong
May 12, 2017

X = csc ⁡ 2 θ − 1 csc ⁡ 2 θ Note that csc ⁡ θ = 1 sin ⁡ θ = 1 sin ⁡ 2 θ − 1 1 sin ⁡ 2 θ Multiplying up and down by sin ⁡ 2 θ = 1 − sin ⁡ 2 θ Note that sin ⁡ 2 θ + cos ⁡ 2 θ = 1 = cos ⁡ 2 θ \begin{aligned} X & = \frac {{\color{#3D99F6} \csc^2 \theta}-1}{\color{#3D99F6} \csc^2 \theta} & \small \color{#3D99F6} \text{Note that } \csc \theta = \frac 1{\sin \theta} \\ & = \frac {\frac 1{\sin^2 \theta}-1}{\frac 1{\sin^2 \theta}} & \small \color{#3D99F6} \text{Multiplying up and down by }\sin^2 \theta \\ & = 1-\sin^2 \theta & \small \color{#3D99F6} \text{Note that } \sin^2 \theta + \cos^2 \theta = 1 \\ & = \boxed{\cos^2 \theta} \end{aligned}

2 Helpful 0 Interesting 0 Brilliant 0 Confused

A very good shortcut is used by you when multiplying up and down by s i n 2 θ sin^2\theta

Syed Hamza Khalid - 4 years ago

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