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# verify the identity. cot theta*sec theta = csc theta

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verify the identity. cot theta*sec theta = csc theta

Jul 16, 2014

#1
+23

Verify cot(x) * sec(x) = csc(x)

cot(x) = 1/tan(x)

tan(x) = sin(x)/cos(x)

So cot(x) = 1/ sin(x) * cos(x)

And sec(x) = 1/cos(x)

So the whole equation is:

1/ sin(x) * cos(x) * 1/cos(x) =csc(x)

The cos(x)  and 1/cos(x) cancel out:

1/sin(x) = csc(x)

csc(x) is the same as 1/sin(x):

csc(x) = csc(x)

:)

Jul 16, 2014

#1
+23

Verify cot(x) * sec(x) = csc(x)

cot(x) = 1/tan(x)

tan(x) = sin(x)/cos(x)

So cot(x) = 1/ sin(x) * cos(x)

And sec(x) = 1/cos(x)

So the whole equation is:

1/ sin(x) * cos(x) * 1/cos(x) =csc(x)

The cos(x)  and 1/cos(x) cancel out:

1/sin(x) = csc(x)

csc(x) is the same as 1/sin(x):

csc(x) = csc(x)

:)

kitty<3 Jul 16, 2014
#2
+8

Thanks Kitty,

Kitty has done it the usual traditional way (the way that Iwould normally do it) - I just thought I would take a look at a more basic method.

$$\\cot\;\theta=\frac{1}{tan\theta}=\frac{adj}{opp}\qquad \mbox{pos in 1st and 3rd quads}\\\\ sec\;\theta=\frac{1}{cos\theta}=\frac{hyp}{adj}\qquad \mbox{pos in 1st and 4th quads}\\\\ cosec\;\theta=\frac{1}{sin\theta}=\frac{hyp}{opp}\qquad \mbox{pos in 1st and 2nd quads}\\\\ \cot\theta\times sec\;\theta=\frac{adj}{opp}\times\frac{hyp}{adj}=\frac{hyp}{opp}=cosec\;\theta$$

Now, this obviously works in the first quadrant where everything is positive but what about in the other quadrants?

2nd quad - x - = +  true

3rd quad  + x - = -  true

4th quad - x + = -  true

Jul 17, 2014