Which of the following are trigonometric identities, I have narrowed it down to two -
(cscx + cotx)^2 = 1
OR
sin^2(x)sec^2(x) + 1 = cot^2(x)csc^2(x)
Before I go through the effort of proving a trig identity, I usually test it with some arbitrary value of an angle
Do not use one of the special angles, such as 30° or 45°, but pick something like 23°
for your first one:
LS = (csc 23° + cot 23°)^2
= (2.559 + 2.3558)^2
which is clearly not equal to 1
so the first one is not an identity.
Do the same with the second equation.
Both of the given statements are trigonometric identities. However, it seems you are only asked to choose one of them. Let's analyze both statements to determine which one is correct.
Statement 1: (cscx + cotx)^2 = 1
To verify if this is a trigonometric identity, we can expand the left side using basic trigonometric definitions and simplify:
(cscx + cotx)^2 = csc^2(x) + 2cscxcotx + cot^2(x)
= (1/sin^2(x)) + 2(1/sinx)(cosx/sinx) + (cos^2(x)/sin^2(x))
= 1/sin^2(x) + 2cosx/sin^2(x) + cos^2(x)/sin^2(x)
Combining the fractions under a common denominator:
= (1 + 2cosx + cos^2(x))/sin^2(x)
On the other hand, we know the identity sin^2(x) + cos^2(x) = 1, so we can substitute this into the numerator:
= (sin^2(x) + 2cosx + cos^2(x))/sin^2(x)
= (1 + 2cosx)/sin^2(x)
Comparing the result with the right side of the equation, which is 1, we see that they are not equal. Therefore, Statement 1 is not a trigonometric identity.
Statement 2: sin^2(x)sec^2(x) + 1 = cot^2(x)csc^2(x)
To verify this statement, we can manipulate it using basic trigonometric identities:
sin^2(x)sec^2(x) + 1 = cot^2(x)csc^2(x)
sin^2(x)(1/cos^2(x)) + 1 = (cos^2(x)/sin^2(x))(1/sin^2(x))
sin^2(x)/cos^2(x) + 1 = cos^2(x)/sin^2(x)
Next, we know the identity sin^2(x) + cos^2(x) = 1, which means sin^2(x) = 1 - cos^2(x). Substituting this into the equation:
(1 - cos^2(x))/cos^2(x) + 1 = cos^2(x)/(1 - cos^2(x))
Now, multiply both sides of the equation by cos^2(x)(1 - cos^2(x)):
(1 - cos^2(x))(1 - cos^2(x)) + cos^2(x)(1 - cos^2(x)) = cos^2(x)cos^2(x)
Expanding this:
(1 - 2cos^2(x) + cos^4(x)) + (cos^2(x) - cos^4(x)) = cos^4(x)
Simplifying:
1 - cos^2(x) + cos^4(x) + cos^2(x) - cos^4(x) = cos^4(x)
1 = cos^4(x)
From this equation, we can see that both sides are indeed equal. Therefore, Statement 2 is a trigonometric identity.
To determine if either of the equations given is a trigonometric identity, we need to verify if they hold true for all values of x.
Let's start with the first equation: (cscx + cotx)^2 = 1.
To confirm this identity, we can start by expanding the left side of the equation and simplifying it:
(cscx + cotx)^2 = (1/sinx + cosx/sinx)^2
= (1 + cosx/sinx)^2
= [(sinx + cosx)/sinx]^2
= (sin^2x + 2sinxcosx + cos^2x) / sin^2x
This expanded form is not equal to 1 for all values of x. Therefore, (cscx + cotx)^2 = 1 is not a trigonometric identity.
Now, let's consider the second equation: sin^2(x)sec^2(x) + 1 = cot^2(x)csc^2(x).
Again, we will expand the left side and simplify it:
sin^2(x)sec^2(x) + 1 = sin^2(x) * (1/cos^2x) + 1
= sin^2(x)/cos^2(x) + 1
= tan^2(x) + 1
Using the pythagorean identity, tan^2(x) + 1 = sec^2(x). Therefore, sin^2(x)sec^2(x) + 1 simplifies to sec^2(x), which is equal to cot^2(x)csc^2(x).
Since the expanded form of the second equation is equal to cot^2(x)csc^2(x) and holds true for all values of x, we can conclude that sin^2(x)sec^2(x) + 1 = cot^2(x)csc^2(x) is a trigonometric identity.