It's A Very Useful Identity For Integration

Geometry Level 2

1 + cos ⁡ 2 θ 1 − cos ⁡ 2 θ \large \dfrac{1 + \cos 2\theta}{1-\cos 2\theta}

Simplify the trigonometric function above.

cot ⁡ 2 θ \cot^2\theta csc ⁡ 2 θ \csc^2\theta sin ⁡ 2 θ \sin^2\theta tan ⁡ 2 θ \tan^2\theta

Pi Han Goh by Pi Han Goh , 30, Malaysia

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4 solutions

Relevant wiki: Proving Trigonometric Identites

1 + cos ⁡ 2 θ 1 − cos ⁡ 2 θ = 1 + 2 cos ⁡ 2 θ − 1 1 − 1 + 2 sin ⁡ 2 θ = 2 cos ⁡ 2 θ 2 sin ⁡ 2 θ = cot ⁡ 2 θ \begin{aligned} \frac{1+\cos 2\theta}{1 - \cos 2\theta} & = \frac{1+2 \cos^2 \theta - 1}{1-1+2 \sin^2 \theta} \\ & = \frac{2\cos^2 \theta}{2 \sin^2 \theta} \\ & = \boxed{\cot^2 \theta} \end{aligned}

16 Helpful 0 Interesting 0 Brilliant 0 Confused

Wonderful simplicity.

A Former Brilliant Member - 5 years ago

Same solution, great!!

Novril Razenda - 4 years, 11 months ago

Same method, just a longer approach.

Mehul Arora - 4 years, 11 months ago

1 + cos ⁡ ( 2 θ ) 2 ⋅ 2 1 − cos ⁡ ( 2 θ ) \frac{1+\cos(2\theta)}{2} \cdot \frac{2}{1-\cos(2\theta)} = cos ⁡ 2 ( θ ) sin ⁡ 2 ( θ ) = cot ⁡ 2 ( θ ) = \frac{\cos^2(\theta)}{\sin^2(\theta)}=\cot^2(\theta)

6 Helpful 0 Interesting 0 Brilliant 0 Confused

cot = cos/sin. Look at what you have written ;)

JPaytheboss . - 5 years ago
Novril Razenda
Jul 1, 2016

Relevant wiki: Proving Trigonometric Identities - Basic

1 + cos ⁡ 2 θ 1 − cos ⁡ 2 θ = s i n 2 θ + c o s 2 θ + cos ⁡ 2 θ − s i n 2 θ s i n 2 θ + c o s 2 θ − cos ⁡ 2 θ + s i n 2 θ = 2 cos ⁡ 2 θ 2 sin ⁡ 2 θ = cot ⁡ 2 θ \begin{aligned} \frac{1+\cos 2\theta}{1 - \cos 2\theta} & = \frac{sin^2\theta+cos^2\theta+ \cos^2 \theta - sin^2\theta}{sin^2\theta+cos^2\theta- \cos^2 \theta +sin^2\theta } \\ & = \frac{2\cos^2 \theta}{2 \sin^2 \theta} \\ & = \boxed{\cot^2 \theta} \end{aligned}

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Very nice and simple!

Pi Han Goh - 4 years, 11 months ago

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@Pi Han Goh Thanks :)

Novril Razenda - 4 years, 11 months ago

Use the double angle identity for cos ⁡ 2 θ \cos 2 \theta and simplify. You get the answer as cot ⁡ 2 θ \cot^2 \theta .

1 Helpful 0 Interesting 0 Brilliant 0 Confused

@Svatejas Shivakumar , since I solved it the same way, I would urge you to post a complete solution instead of hinting the reader about which direction to go ;)

Mehul Arora - 4 years, 11 months ago

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I solved it the same way as chew seong cheong sir. So no need to post the solution.

A Former Brilliant Member - 4 years, 11 months ago

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