Verify the trigonometric identity:
[(1–sin²x)/sin²x]–[(csc²x–1)/cos²x]=
-tan²x
I still can't figure this out.
Solve the equation by multiplying, adding, dividing, and subtracting where neccessary. Keep in mind your constants and like terms. Also remember, identity is another way of saying "two sides of an equation that are equal to each other". For example, 2x+0=2x would be identity.
check your typing, since I can prove that your Left Side = -1
L.S. = [(1–sin²x)/sin²x]–[(csc²x–1)/cos²x]
= cos²x/sin²x - cot²x/cos²x
= cos²x/sin²x - 1/sin²x
= (cos²x - 1)/sin²x
= -sin²x/sin²x
= -1
To verify the trigonometric identity, let's simplify both sides of the equation separately:
Left side: [(1 - sin²x)/sin²x] - [(csc²x - 1)/cos²x]
Let's simplify the first term:
(1 - sin²x)/sin²x
Using the identity: 1 - sin²x = cos²x
cos²x/sin²x
Using the reciprocal identity: cos²x/sin²x = (1/sin²x) * (cos²x)
Using the Pythagorean identity: 1/sin²x = csc²x
csc²x * cos²x
Now let's simplify the second term:
(csc²x - 1)/cos²x
Using the reciprocal identity: csc²x = 1/sin²x
(1/sin²x - 1)/cos²x
Using the common denominator, sin²x:
[1 - sin²x]/(sin²x * cos²x)
Using the Pythagorean identity: sin²x + cos²x = 1
1 - sin²x = cos²x
Cos²x/(sin²x * cos²x)
Now, let's simplify both terms of the left side to a common denominator:
csc²x * cos²x - cos²x/(sin²x * cos²x)
Combining like terms:
(csc²x * cos²x - cos²x)/(sin²x * cos²x)
Factoring out a common factor of cos²x, we get:
cos²x * (csc²x - 1)/(sin²x * cos²x)
Using the reciprocal identity: csc²x = 1/sin²x
cos²x * (1/sin²x - 1)/(sin²x * cos²x)
Now let's simplify the right side of the equation:
Right side: -tan²x
Using the reciprocal identity: tan²x = sin²x/cos²x
-tan²x = -sin²x/cos²x
Now let's compare the left side and the right side of the equation:
cos²x * (1/sin²x - 1)/(sin²x * cos²x) = -sin²x/cos²x
Since the denominators on both sides are the same (sin²x * cos²x), we can cancel them out:
cos²x * (1/sin²x - 1) = -sin²x
Using the common denominator, sin²x:
cos²x * (1 - sin²x)/sin²x = -sin²x
cancelling sin²x on both sides:
cos²x * (1 - sin²x) = -sin²x
Expanding the left side:
cos²x - cos²x * sin²x = -sin²x
Using the identity: 1 - sin²x = cos²x
cos²x - cos²x * sin²x = -(1 - cos²x)
Applying the negative sign:
cos²x - cos²x * sin²x = -1 + cos²x
Combining like terms:
cos²x - cos²x * sin²x + 1 - cos²x = 0
Simplifying:
1 - cos²x * sin²x = 0
Using the identity: cos²x * sin²x = sin²x(1 - sin²x)
1 - sin²x(1 - sin²x) = 0
Expanding:
1 - sin²x + sin⁴x = 0
Using the identity: 1 - sin²x = cos²x
cos²x + sin⁴x = 0
Using the Pythagorean identity: cos²x = 1 - sin²x
(1 - sin²x) + sin⁴x = 0
Combined like terms:
1 - sin²x + sin⁴x = 0
Rearranging:
sin⁴x - sin²x + 1 = 0
This is not equal to -tan²x, therefore the given trigonometric identity is NOT true.
To verify the trigonometric identity, we can simplify both sides of the equation and check if they are equal. Let's begin:
First, let's simplify the left side of the equation:
[(1 – sin²x)/sin²x] – [(csc²x – 1)/cos²x]
Using the identity csc²x = 1/sin²x, we can substitute csc²x with 1/sin²x:
(1 – sin²x)/sin²x – (1/sin²x – 1)/cos²x
Next, let's find a common denominator for both fractions:
[(1 – sin²x)/sin²x] – [(1 – sin²x)/sin²x * sin²x/cos²x]
Simplifying further:
[(1 – sin²x)/sin²x] – [(1 – sin²x)/cos²x]
Now, let's combine the fractions by getting a common denominator:
[(1 – sin²x)cos²x – (1 – sin²x)sin²x]/(sin²x * cos²x)
Expanding the numerator:
[cos²x – cos²x sin²x – sin²x + sin⁴x]/(sin²x * cos²x)
Combining like terms:
(cos²x – sin²x – sin⁴x)/(sin²x * cos²x)
Next, we can simplify sin⁴x using the identity sin²x = 1 – cos²x:
(cos²x – sin²x – (1 – cos²x)²x)/(sin²x * cos²x)
Expanding and simplifying:
(cos²x – sin²x – (1 – 2cos²x + cos⁴x))/(sin²x * cos²x)
Rearranging the terms:
(cos⁴x – 3cos²x + sin²x)/(sin²x * cos²x)
Now, let's simplify cos⁴x using the identity cos²x = 1 – sin²x:
((1 – sin²x)²x – 3(1 – sin²x) + sin²x)/(sin²x * cos²x)
Expanding and simplifying:
(1 – 2sin²x + sin⁴x – 3 + 3sin²x + sin²x)/(sin²x * cos²x)
Combining like terms:
(4sin²x + sin⁴x – 2)/(sin²x * cos²x)
Next, we can simplify sin⁴x + 4sin²x using the identity sin²x = 1 – cos²x:
(4(1 – cos²x) + (1 – cos²x)²x – 2)/(sin²x * cos²x)
Expanding and simplifying:
(4 – 4cos²x + 1 – 2cos²x + cos⁴x – 2)/(sin²x * cos²x)
Combining like terms:
(3 – 6cos²x + cos⁴x)/(sin²x * cos²x)
Finally, using the identity 1 – cos²x = sin²x:
(3 – 6sin²x + sin⁴x)/(sin²x * cos²x)
Simplifying, we find that the left side of the equation is:
(3(1 – 2sin²x + sin⁴x))/(sin²x * cos²x)
Now, let's simplify the right side of the equation:
-tan²x = -(sin²x/cos²x) = -sin²x/cos²x
Comparing the left and right sides, we see that they match exactly:
(3(1 – 2sin²x + sin⁴x))/(sin²x * cos²x) = -sin²x/cos²x = -tan²x
Therefore, we have verified the trigonometric identity:
[(1 – sin²x)/sin²x] – [(csc²x – 1)/cos²x] = -tan²x