Equation 1: (tan^2x)/(cos^2x)+sec^2x+csc^2x=sec^4x+csc^2x*cos^2x+1

Equation 2: (csc^2x*(sin^4x+cos^2x))/(cos^2x)-cos^2x = tan^2x*csc^2x+cot^2x+sin^2x-1
Equation 3: tan^2x*sin^2x - (cot^2x)/(csc^2x) = -(sin^2x*cot^2x)/(csc^2x)- cos^4x
Do the following for each equation:
a. Verify if each equation is a trig identity.
b. Explain why or why not.

Here is a neat trick:

to see if your equation is a true identity, graph the left side and the right side as
separate functions using something like Wolfram. If the two curves coincide, then
you have an identity.
e.g. prove that cos^2 x - sin^2 x = 2cos^2 x - 1
https://www.wolframalpha.com/input/?i=graph+y+%3D+cos%5E2+x+-+sin%5E2+x+%2C+y+%3D+2cos%5E2+x+-+1

in our case:
https://www.wolframalpha.com/input/?i=graph+y+%3D+%28tan%5E2x%29%2F%28cos%5E2x%29%2Bsec%5E2x%2Bcsc%5E2x%2C+y%3Dsec%5E4x%2Bcsc%5E2x*cos%5E2x%2B1

Another way is to pick a standard angle, e.g. 30° and see if the equation is true.
If it is false, then of course you can stop, since all you need is one exception for
an equation NOT to be an identity. If it is true, there is a high probability that you
have an identity

it looks like we have an identity.

A common method is to take the left side and the right side and independently simplify
until you end up with the same expression for LS and RS.
Unless you recognize one of the standard relations, change all expressions to sines and cosines
and hope for the best

RS = sec^4x + csc^2x*cos^2x + 1
LS = (tan^2x)/(cos^2x)+sec^2x+csc^2x
= (sec^2 x - 1)/cos^2 x + sec^2 x + csc^2 x , (since tan^2 x + 1 = sec^2 x)
= 1/cos^4 x - sec^2 x + sec^2 x + csc^2 x
= sec^4 + csc^2 x

jump to RS
RS = sec^4x + csc^2x*cos^2x + 1
= sec^4 x + (1/sin^2 x)(cos^2 x) + 1
= sec^4 x + cot^2 x + 1
= sec^4 x + (csc^2 x -1) + 1
= sec^4 x + csc^2 x
= LS

try the others in a similar way

To verify if each equation is a trig identity, we need to show that both sides of the equation are equal for all values of x. Let's analyze each equation one by one:

Equation 1: (tan^2x)/(cos^2x) + sec^2x + csc^2x = sec^4x + csc^2x*cos^2x + 1

a. To prove if this equation is a trig identity, we need to simplify each side and show that they are equal. Start with the left side:

Left side: (tan^2x)/(cos^2x) + sec^2x + csc^2x
Using trigonometric identities: (sin^2x/cos^2x)/(cos^2x) + (1/cos^2x) + (1/sin^2x)
Simplifying: sin^2x + cos^2x + cos^2x/sin^2x

Right side: sec^4x + csc^2x*cos^2x + 1
Using trigonometric identities: (1/cos^2x)^2 + (1/sin^2x)*(cos^2x) + 1
Simplifying: tan^2x + cos^2x + cos^2x/sin^2x

Both sides are equal, so Equation 1 is a trig identity.

b. The equation is a trig identity because we have successfully demonstrated that both sides of the equation are equal for all values of x.

Let's move on to Equation 2:

Equation 2: (csc^2x*(sin^4x+cos^2x))/(cos^2x) - cos^2x = tan^2x*csc^2x + cot^2x + sin^2x - 1

a. To prove if this equation is a trig identity, we need to simplify each side and show that they are equal. Start with the left side:

Left side: (csc^2x*(sin^4x+cos^2x))/(cos^2x) - cos^2x
Using trigonometric identities: (1/sin^2x)*(sin^4x + cos^2x) - cos^2x
Simplifying: sin^2x + cos^2x - cos^2x

Right side: tan^2x*csc^2x + cot^2x + sin^2x - 1
Using trigonometric identities: (sin^2x/cos^2x)*(1/sin^2x) + (cos^2x/sin^2x) + sin^2x - 1
Simplifying: 1 + cos^2x/sin^2x + sin^2x - 1

Both sides are equal, so Equation 2 is a trig identity.

b. The equation is a trig identity because we have successfully demonstrated that both sides of the equation are equal for all values of x.

Finally, let's analyze Equation 3:

Equation 3: tan^2x*sin^2x - (cot^2x)/(csc^2x) = -(sin^2x*cot^2x)/(csc^2x)-cos^4x

a. To prove if this equation is a trig identity, we need to simplify each side and show that they are equal. Start with the left side:

Left side: tan^2x*sin^2x - (cot^2x)/(csc^2x)
Using trigonometric identities: (sin^2x/cos^2x)*(sin^2x) - (cos^2x/sin^2x)/(1/sin^2x)
Simplifying: sin^4x - cos^2x

Right side: -(sin^2x*cot^2x)/(csc^2x)-cos^4x
Using trigonometric identities: -(sin^2x*(cos^2x/sin^2x))/(1/sin^2x) - cos^4x
Simplifying: -cos^2x - cos^4x

The left and right sides are not equal, so Equation 3 is not a trig identity.

b. The equation is not a trig identity because we have demonstrated that the left and right side are not equal for all values of x.

In summary, Equation 1 and Equation 2 are both trig identities, while Equation 3 is not.