Prove: (tan y + Cot y) sin y cos y = 1

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In summary, the conversation discusses the need to prove an equation involving trigonometric functions. The person has simplified the equation to (tan y + Cot y) sin y cos y = 1 and has gotten stuck. They suggest using the ratios sin theta = o/h and cos theta = a/h to simplify the equation and eventually solve it. After simplifying, they realize that the equation can be rearranged to equal 1.
  • #1
Jasonp914
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Hi, I need to prove:

(tan y + Cot y) sin y cos y = 1

I've gotten this far and got stuck:

(tany + cos y) sin y cos y changed Cot to reciprical
--------sin y

(sin y tany + cosy) sin y cos y common demonator
--------siny

(sin y tany + cos y) cosy multipyied by sin y

then i could distribute

cos y sin y tan y + cos(sqared) y but how would that equal 1?

thanks for your time.
 
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  • #2
It should help if you plug in

[tex]\tan x = \frac{\sin x}{\cos x}[/itex]
 
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  • #3
just a suggestion of my own, but try expressing those ratios literally in terms of fractions (ie: sin theta = o/h)...simplify the equation now and see if you notice anything familiar.
 
  • #4
o got it thank you!
 

1. How do you prove the equation (tan y + Cot y) sin y cos y = 1?

To prove this equation, we will use the trigonometric identities of tangent and cotangent: tan y = sin y / cos y and cot y = cos y / sin y. Substituting these identities into the original equation, we get (sin y / cos y + cos y / sin y) sin y cos y = 1. Simplifying this expression, we get sin^2 y + cos^2 y = 1, which is a well-known trigonometric identity and proves the original equation.

2. Can you provide a visual representation of this equation?

Yes, we can represent this equation using a unit circle. If we draw a right triangle with sides of length sin y, cos y, and 1, we can see that the sum of the tangent and cotangent of angle y equals 1. This is because tan y is the opposite side over the adjacent side, and cot y is the adjacent side over the opposite side, and together they form a right triangle with a hypotenuse of 1.

3. What is the significance of this equation in trigonometry?

This equation is significant because it shows the relationship between the tangent, cotangent, sine, and cosine functions. It also demonstrates a fundamental trigonometric identity that is used in many other equations and proofs.

4. Can this equation be used to solve trigonometric equations?

Yes, this equation can be used to solve other trigonometric equations by rearranging the terms and substituting in known values. For example, we can use it to solve for y if we know the values of sin y, cos y, and tan y.

5. Are there any limitations to this equation?

One limitation of this equation is that it only applies to right triangles. It also assumes that y is a non-zero angle. Additionally, this equation may not hold true for values of y that make sin y or cos y equal to 0, as dividing by 0 is undefined.

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