Question

Simplify and write in terms of sine.
(csc^2Ɵ - cot^2Ɵ) sec^2Ɵ

Answer 1

Note that \(\csc^2\theta-\cot^2\theta = \dfrac{1}{\sin^2\theta}-\dfrac{\cos^2\theta}{\sin^2\theta} = \dfrac{1-\cos^2\theta}{\sin^2\theta}\), right?

\(1-\cos^2\theta\) looks like it can be simplified... maybe it has something to do with \(\sin^2\theta+\cos^2\theta=1\)?

Answer 2

\[1-\cos^2\theta = \sin^2\theta \] @micahwood50

Answer 3

Right, so you can replace that to \(\sin^2\theta\), then simplify ahead

Answer 4

You should see that \(\csc^2\theta-\cot^2\theta\) is equal to one.

Answer 5

ohhh I get it so it will be \[\frac{ \sin^2\theta }{ \sin^2\theta }\]

@micahwood50

Answer 6

Right, hence 1

Answer 7

So you have this: \((\csc^2\theta-\cot^2\theta)\sec^2\theta ~\Rightarrow~(1)\sec^2\theta~\Rightarrow~?\)

Answer 8

So now I have to write \[\sec^2\theta \] in terms of sine? @micahwood50

Answer 9

Isn't \[\sec^2 \theta = \frac{ 1 }{ \cos^2 \theta }\] @micahwood50

Answer 10

oh right, i forgot about writing in term of sine. just rewrite \(\dfrac{1}{\cos^2\theta}\).

Again, \(\sin^2\theta+cos^2\theta=1\)