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Prove the identity (1 - sin x)/(cos x) = (cos x)/(1 + sin x).  Are there any x for which the equation does not hold?

The only integers that satisfy x^2=2y^2 are x=y=0

Hi my friend

120/(2-12)=120/-1=-12

If x,y, and z were other integers then it would not satisfy the equation. x=y=z=0

Haha nice

The formula for diagonals is n(n-3)/2 so the answer is 20*17/2which is 10*17=170

Simplify (2*sqrt(3) + sqrt(10))*(7*sqrt(6) - 4)*(2*sqrt(3) - sqrt(10)).

8+(14*sqrt(6))

"A glass is half full of pure alcohol and another glass twice the volume is one third full.  Both glasses are then filled to the top with water and mixed together.  What is the fraction of alcohol in the resulting mixture?"

Edited to correct mistake.  Thnaks to EP for pointing this out.

hi dimitris 

I found it thank you.

1/3 full in twice the volume is the same as 2/3 full in SAME volume 

    1/2  + 2/3 = 7/6   100% alcohol

then  1/2 + 4/3 water is added   = 11/6 water    

      7/6 alcohol  and  11/6  water

       7/6  /( 7/6 + 11/6) =   7/6 /(3) = 7/18 = 38.8% alcohol

Example :  container A is  12 oz         6 oz of alcohol        + 6 oz of water

                  container B is  24 oz         8 oz of alcohol        + 16 oz of water

      Total :                          36 oz        14 oz of alcohol

                                             14/36 = 7/18 is alcohol = 38.89% alcohol

thank you!

by which theorem? 

A heptagon has 7 sides.

A septagon has 7 sides.

"Heptagon" and "septagon" both describe the same figure.

A nonagon has 9 sides.

You know that EB + CF = BC.

I believe that the first (largest) blue areas of the triangle form 3/4^ths of the complete triangle.

The next blue area forms 3/4^ths of 1/4^th of 1/4^th of the triangle.

The next blue area forms 3/4^ths of 1/4^th of 1/4^th of 1/4^th of 1/4^th of the triangle.

Etc.

(3/4)  +  (3/4)(1/4)²  +  (3/4)(1/4)⁴  +  (3/4)(1/4)⁶  +  ...

This forms a geometric sequence with  first term  =  3/4  and with  common ratio  =  1/16:

Sum  =  (3/4) / [ 1 - 1/16 ]  =  (3/4) / (15/16)  =  4/5.

120 / (2-12) = 120 / -10 = -12.

1.6 mm = .0016 m

.0016 m / (200x 10^-12 m/layer) = 8 000 000 layers = 8 x 10⁶  layers

120/ ( 2-12) =    120/ ( -10) = - 12     (edited)

As follows:

Edit:  The third line in my derivation should say AD² not AO².  The numbers are correct though.

1)  To find the radius of the circle, we need to find the distance from (0,0) to (-4,,2).

      Distance formula:  d  =  sqrt( (x₂ - x₁)² + (y₂ - y₁)² )

                                        =  sqrt( (-4 - 0)² + (2 - 0)² )

                                        =  sqrt( 16 + 4)

                                        =  sqrt( 20 )  

                                        =  sqrt( 4 · 5 )

                                        =  sqrt( 4 ) · sqrt( 5 )

                                        =  2 · sqrt( 5 )

2)  If the equation for a circle is in this form:  (x - h)² + (y - k)²  =  r²

     then the center is at (h, k) and the radius is r.

     (x - 1)² + (y + 7)²  =  29     --->     center  =  (1,-7)     radius  =  sqrt( 29 )

Solve for t:                                           u  =  7t + 30

subtract 30 from both sides:        u - 30  =  7t

divide both sides by 7:          (u - 30) / 7  =  t

swithch sides:                                      t  =  (u - 30) / 7

The angle bisector divides the side in the same proportion as the other two sides of the triangle.

So:  BC : AB  =  CP : PA

If  CP  =  x,  then  PA  =  5 - x.

--->   4 : 6  =  x : 5 - x

         4 / 6  =  x / (5 - x)

    4·(5 - x)  =  6·x

      20 - 4x  =  6x

             20  =  10x

               x  =  2           --->     CP  =  2

          5 - x  =  3           --->     PA  =  3