Proving the Identity: tan(theta)sec(theta)-1 / tan(theta)-sec(theta)+1 = sin(theta)/cos(theta)

To prove this trigonometric identity, we will manipulate the left-hand side (LHS) of the equation and simplify it until it is equal to the right-hand side (RHS), step by step.

Step 1: Simplifying the LHS

Let's start by simplifying the expression on the LHS of the equation:

tan(theta)sec(theta)-1 / tan(theta)-sec(theta)+1

We can simplify this expression by multiplying both the numerator and denominator by the conjugate of the denominator, which is tan(theta)+sec(theta).

(tan(theta)sec(theta)-1) * (tan(theta)+sec(theta)) / (tan(theta)-sec(theta)+1) * (tan(theta)+sec(theta))

Step 2: Expanding the Numerator

Now, we expand the numerator:

(tan(theta)sec(theta) * tan(theta)) + (tan(theta)sec(theta) * sec(theta)) - (tan(theta) + sec(theta))

Step 3: Simplifying the Numerator

Next, we simplify the numerator further:

sin(theta) + 1 - (tan(theta) + sec(theta))

Step 4: Simplifying the Denominator

Now, let's simplify the denominator:

(tan(theta))^2 - (sec(theta))^2 + 1

Using the Pythagorean identities, we know that (tan(theta))^2 + 1 = (sec(theta))^2. Therefore, we can substitute (sec(theta))^2 - 1 in place of (tan(theta))^2:

(sec(theta))^2 - (sec(theta))^2 + 1

This simplifies to just 1.

Step 5: Final Simplification

Finally, we can simplify the entire expression by substituting the simplified numerator and denominator back into the original equation:

(sin(theta) + 1 - (tan(theta) + sec(theta))) / 1

Since any number divided by 1 is equal to the number itself, we are left with:

sin(theta) + 1 - (tan(theta) + sec(theta))

This can be further simplified to:

sin(theta) - tan(theta) - sec(theta) + 1

Step 6: Comparing with the RHS

Now, let's compare this simplified expression with the RHS of the equation, which is sin(theta)/cos(theta).

sin(theta)/cos(theta)

Using trigonometric identities, we know that sin(theta)/cos(theta) is equivalent to tan(theta). Therefore, the RHS is equal to tan(theta).

Conclusion:

We have shown that the LHS of the equation, sin(theta) - tan(theta) - sec(theta) + 1, is equal to the RHS, tan(theta). Therefore, we have successfully proven the identity: tan(theta)sec(theta)-1 / tan(theta)-sec(theta)+1 = sin(theta)/cos(theta).