Answer:
the answer is the letter a) -sin x
Step-by-step explanation:
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=11−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)
Answer:
[tex]-cos \ x[/tex]
Step-by-step explanation:
First of all, we must have to understand what is the described expression in the paragraph
"sine of x to the second power minus one divided by cosine of negative x"
In this sentence, we need to identify what are the elements and operations involved in the expression.
In the sentence appears ""to the second power", "minus" and "divided by" (highlighted)
"sine of x to the second power minus one divided by cosine of negative x"
Therefore, the expression must has three operations:
Now, we can identify what are the elements: "sine of x", "one" and "cosine of negative x"
Therefore, the expression is:
[tex]\frac{(sin\ x)^{2}-1}{cos(-x)}[/tex]
In order to find the simplified expression, we must have to apply these trigonometric identities:
Applying the first and third identities, we have:
[tex]\frac{(sin\ x)^{2}-1}{cos(-x)}=\frac{sin\x^{2}x-1}{cos\ x}[/tex]
From the second trigonometric identity, we have:
[tex]cos\x^{2}x=\ 1-sin\x^{2}x[/tex]
Now, multiplying by -1 in both sides:
[tex](-1)(cos\x^{2}x)=(-1)(1-\ sin\x^{2}x)[/tex]
In the left side, multiplying by -1 the sign of the expression changes:
[tex](-1)(cos\x^{2}x)=-cos\x^{2}x[/tex]
In the right side, multiplying by -1 changes the order of the substraction:
[tex](-1)(1-\ sin\x^{2}x)=\ sin\x^{2}x-1[/tex]
Putting all together:
[tex]-cos\x^{2}x=\ sin\x^{2}x-1[/tex]
Now, replacing values we have:
[tex]\frac{sin\x^{2}x-1}{cos\ x}=\frac{-cos\x^{2}x}{cos\ x}=-\frac{cos\x^{2}x}{cos\ x}[/tex]
Finally, the property of the first trigonometric identity (property of exponentiation) can be apply in this case:
[tex]-\frac{cos\x^{2}x}{cos\ x}=-\frac{(cos\ x)^{2}}{cos\ x}=-cos\ x[/tex]
