TRIG Identities and Equations

These are the problems I have trouble with while doing a practice on this topic.

1. Factor the expression and use fundamental identities to simplify:

cos^(2)x csc^(2)x - cos^(2)x

2. Find all solutions in the interval [0 , 2pi)

tan^(2)θ = (-3/2)secθ

3. Find an expression that completes the identity

[2sin^(2)x + cos(2x)] / csc(x) =

It would be helpful if u could explain step by step. I'm not just looking for answers. Thanks

Re: TRIG Identities and Equations

In the first one, take out $\displaystyle \cos^2x $ as a factor, what do you get?

For the second, create a quadratic using $\displaystyle \tan^2\theta = \sec^2\theta -1$ what do you get?

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Re: TRIG Identities and Equations

Re: TRIG Identities and Equations

In 3 use the identity cos2x=1-2sin^(2)x

Re: TRIG Identities and Equations

Hello, Crysland!

Quote:

$\displaystyle \text{3. Find an expression that completes the identity,}$

. . $\displaystyle \frac{2\sin^2\!x + \cos2x}{\csc x}\:=\:\_\_\_$

Replace $\displaystyle \cos2x$ with $\displaystyle \cos^2\!x - \sin^2\!x$

$\displaystyle \frac{2\sin^2\!x + \cos2x}{\csc x} \;=\;\frac{2\sin^2\!x + (\cos^2\!x - \sin^2\!x)}{\csc x}$

. . . . . . . . . . . $\displaystyle =\; \frac{\overbrace{\sin^2\!x + \cos^2\!x}^{\text{This is 1}}}{\csc x}$

. . . . . . . . . . . $\displaystyle =\; \frac{1}{\csc x}$

. . . . . . . . . . . $\displaystyle =\;\sin x $

Re: TRIG Identities and Equations

thanks so much. I'm still stuck with #1.

I got 2 answers, cos^2x*cot^2x and cos^4x/sin^2x

but don't know which one is correct and simplified.

Re: TRIG Identities and Equations

Both your answers equal one another (because cotx=cosx/sinx)

Re: TRIG Identities and Equations

Re: TRIG Identities and Equations