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)
:)
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)
:)
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