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The answer depends on how the ratios are defined. In some cases tan is DEFINED as the ratio of sine and cosine rather than from the angle in a right angled triangle.


If the trig ratios were defined in terms of a right angled triangle, thensine is the ratio of the opposite side to the hypotenuse,

cosine is the ratio of the adjacent side to the hypotenuse,

and tangent is the ratio of the opposite side to the adjacent side.

It is then easy to see that sin/cos = (opp/hyp)/(adj/hyp) = opp/adj = tan.


If sine and cosine are defined as infinite sums for angles measured in radians, ie

sin(x) = x - x^3/3! + x^5/5! - x^7/7! + ...

and

cos = 1 - x^2/2! + x^4/4! - x^6/6! + ...

then it is less easy to see tan = sin/cos.

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8y ago

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More answers

The expression tan(theta) sin(theta) / cos(theta) simplifies to sin^2(theta) / cos(theta). In trigonometry, sin^2(theta) is equal to (1 - cos^2(theta)), so the expression can be further simplified to (1 - cos^2(theta)) / cos(theta).

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1y ago
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Q: Why tan theta sin theta divided by cos theta?
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