4. Simplify the expression. (6 points)
sine of x to the second power minus one divided by cosine of negative x


A) -sin x
B) cos x
C) sin x
D) -cos x

5. Find all solutions in the interval [0, 2π). (6 points)
sin2x + sin x = 0



A) x = 0, π, four pi divided by three , five pi divided by three
B) x = 0, π, pi divided by three , two pi divided by three
C) x = 0, π, pi divided by three , five pi divided by three
D) x = 0, π, three pi divided by two


PLEASE HELPP

Respuesta :

4. Simplify the expression.
sine of x to the second power minus one divided by cosine of negative x

(1−sin2(x))/(sin(x)−csc(x))

sin2x+cos2x=1
1−sin2x=cos2x

cos2(x)/(sin(x)−csc(x))
csc(x)=1/sin(x) cos2(x)/(sin(x)− 1/sin(x))= cos2(x)/((sin2(x)− 1)/sin(x)) sin2(x)− 1=-cos2(x) cos2(x)/(( -cos2(x))/sin(x))
=-sin(x)

the answer is the letter a) -sin x

5. Find all solutions in the interval [0, 2π). (6 points)
sin2x + sin x = 0
 using a graphical tool  
the solutions   x1=0 x2=pi x3=3pi/2
the answer is the letter 
D) x = 0, π, three pi divided by two
Ver imagen calculista

Answer with explanation:

Ques 4)

 We are given a trignometric expression by:

     [tex]\dfrac{\sin^2x-1}{\cos (-x)}[/tex]

which could also be written as:

[tex]=\dfrac{-(1-\sin^2x)}{\cos x}[/tex]

since, we know that:

[tex]\cos (-x)=\cos x[/tex]

Also,

[tex]1-\sin^2 x=\cos^2 x[/tex]

Hence, we get:

[tex]\dfrac{\sin^2x-1}{\cos (-x)}=\dfrac{-\cos^2x}{\cos x}\\\\\\\dfrac{\sin^2x-1}{\cos (-x)}=-\cos x[/tex]

            The correct option is:

                        D)   -cos x        

Ques 5)

We are asked to find the solution in the interval [0,2π) of the expression:

        [tex]\sin 2x+\sin x=0[/tex]

This expression could also be written as:

[tex]2\sin x\cos x+\sin x=0\\\\i.e.\\\\\sin x(2\cos x+1)=0\\\\i.e.\\\\Either\ \sin x=0\ or\ 2\cos x+1=0[/tex]

If

[tex]\sin x=0[/tex]

Then the possible values of x are:

[tex]x=0,\pi[/tex]

and if

[tex]2\cos x+1=0\\\\i.e.\\\\\cosx=\dfrac{-1}{2}[/tex]

We know that the cosine function is negative in second and third quadrant and the possible values where x is negative is:

[tex]x=\dfrac{2\pi}{3}\ ,\ x=\dfrac{4\pi}{3}[/tex]

Hence, all the solutions of the given expression that will lie in the given region is:

    [tex]x=0,\ \pi\ ,\ \dfrac{2\pi}{3}\ ,\ \dfrac{4\pi}{3}[/tex]

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