Prove:

1) 1 / sec X - tan X = sec X + tan X

2) cot A + tan A = sec A csc A

3)sec A - 1 / sec A + 1 = 1 - cos A / 1 + cos A

I usually start these with changing everything to sines and cosines after I test them with some angle that I pick

#1. The question should say:
1/(secx - tanx) = secx + tanx
Those brackets are critical, the way you typed it, will not work

LS = 1/(1/cosx - sinx/cosx)
= 1/((1- sinx)/cosx )
= cosx/(1-sinx)

RS = 1/cosx + sinx/cosx
= (1+sinx)/cosx
= (1+sinx)/cosx * (1-sinx)/(1-sinx)
= (1 - sin^2 x)/(cosx(1-sinx)
= cos^2 x/(cosx(1-sinx))
= cosx/(1-sinx)
= LS
that was a tricky one!

#2 this one is easier, try it the way I did the first one

let me see your steps

#3, again, brackets are really important to establish the correct order of operation
(sec A - 1)/(sec A + 1) = (1 - cos A)/(1 + cos A)

I usually start with the more messy looking side
LS = (1/cosA - 1)/(1/cosA + 1)
= [ (1- cosA)/cosA ] / [ (1 + cosA)/cosA ]
= (1- cosA)/cosA * cosA/(1+cosA)
= (1 - cosA)/(1+cosA)
= RS

well, how about that ?

this particular sec/tan example falls out nicely:

1/(secx - tanx) = secx + tanx
1 = (secx+tanx)(secx-tanx)
1 = sec^2(x) - tan^2(x)

one of the fundamental identities, arising immediately from

sin^2 + cos^2 = 1

by dividing by cos^2

Purists will argue that to prove the validity of an identity, that is, prove that the equation is true, by using the rules of equations itself is not valid.

To prove these trigonometric identities, we will use basic trigonometric definitions and properties. Here's how we can approach each proof:

1) To prove 1 / sec X - tan X = sec X + tan X:

Start with the left side of the equation:
1 / sec X - tan X

We know that secant is the reciprocal of cosine:
1 / sec X = cos X

Substituting this into the equation, we have:
cos X - tan X

The tangent of an angle is equal to sin X / cos X:
cos X - sin X / cos X

Finding a common denominator, we get:
(cos X - sin X) / cos X

Now, let's simplify the right side of the equation:
sec X + tan X

Using the definition of secant as the reciprocal of cosine, we have:
1 / cos X + sin X / cos X

Combining the fractions, we get:
(1 + sin X) / cos X

Now the equation becomes:
(cos X - sin X) / cos X = (1 + sin X) / cos X

Since the denominators are the same, we can eliminate them:
cos X - sin X = 1 + sin X

Rearranging the terms, we have:
cos X - sin X - sin X = 1

Combining like terms, we get:
cos X - 2sin X = 1

Now, we can apply the Pythagorean identity for cosine and sine:
cos^2 X + sin^2 X = 1

Rearranging this, we have:
cos^2 X = 1 - sin^2 X

Substituting this into the previous equation, we get:
1 - sin^2 X - 2sin X = 1

Simplifying further, we get:
- sin^2 X - 2sin X = 0

Factoring out sin X, we have:
sin X(-sin X - 2) = 0

From this, we can conclude that either sin X = 0 or -sin X - 2 = 0.

If sin X = 0, then X = 0 or X = π are solutions, which satisfy the original equation.

If -sin X - 2 = 0, then sin X = -2, but this is not a valid solution since the sine function ranges from -1 to 1.

Therefore, the solutions X = 0 and X = π satisfy the equation, and we have proven the identity.

2) To prove cot A + tan A = sec A csc A:

Start with the left side of the equation:
cot A + tan A

The cotangent function is equal to cos A / sin A:
cot A = cos A / sin A

Substituting this into the equation, we have:
cos A / sin A + sin A / cos A

Finding a common denominator, we get:
(cos A cos A + sin A sin A) / (sin A cos A)

Using the Pythagorean identity for cosine and sine, we have:
(1) / (sin A cos A)

Recall that secant is the reciprocal of cosine, and cosecant is the reciprocal of sine:
sec A = 1 / cos A
csc A = 1 / sin A

Substituting these into the equation, we get:
1 / (sin A cos A) = sec A csc A

This shows that the left side of the equation is equal to the right side, proving the identity.

3) To prove sec A - 1 / sec A + 1 = 1 - cos A / 1 + cos A:

Start with the left side of the equation:
sec A - 1 / sec A + 1

Recall that secant is the reciprocal of cosine:
1 / cos A - 1 / (1 / cos A) + 1

Simplifying this expression, we get:
1 / cos A - cos A + 1

Using a common denominator, we have:
1 - cos^2 A + cos A

We can apply the Pythagorean identity for cosine:
1 - (1 - sin^2 A) + cos A

Simplifying further, we obtain:
sin^2 A + cos A

Using the identity sin^2 A + cos^2 A = 1, we replace sin^2 A with 1 - cos^2 A, resulting in:
1 - cos^2 A + cos A

Rearranging this expression, we get:
1 + cos A - cos^2 A

Factoring out a (1 - cos A):
(1 - cos A)(1 + cos A)

Now, let's simplify the right side of the equation:
1 - cos A / 1 + cos A

Applying the rule for dividing fractions, we have:
(1 - cos A)(1 / (1 + cos A))

The expression on the right side is the same as the expression we obtained for the left side.

Therefore, we have proven the identity.