Fractions In Fractions? Too Much!

Geometry Level 2

We know that the identity sin ⁡ x cos ⁡ x = tan ⁡ x \dfrac{\sin x}{\cos x} = \tan x is true. Would the equation still holds true if we replace all the trigonometric functions with their reciprocal counterparts? That is, is the following identity true as well?

csc ⁡ x sec ⁡ x = cot ⁡ x \dfrac{ \csc x}{\sec x} = \cot x

No, it is not true Yes, it is true

Pi Han Goh by Pi Han Goh , 30, Malaysia

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2 solutions

Sam Bealing
Jun 12, 2016

csc ⁡ x sec ⁡ x = 1 sin ⁡ x 1 cos ⁡ x = cos ⁡ x sin ⁡ x = cot ⁡ x \dfrac{\csc{x}}{\sec{x}}=\dfrac{\frac{1}{\sin{x}}}{\frac{1}{\cos{x}}}=\dfrac{\cos{x}}{\sin{x}}=\cot{x}

Yes \color{#20A900}{\boxed{\boxed{\text{Yes}}}}

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Just convert the and denominator in terms of sin ⁡ x \sin x and cos ⁡ x \cos x and simplify. The identity is true.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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