Proving 1/sec theta - tan theta = sin/cos theta

To prove the given equation, we need to simplify the left-hand side and the right-hand side of the equation and show that they are equal.

Simplifying LHS

1/sec theta - tan theta

We know that secant of theta is equal to 1/cosine of theta. Therefore, we can replace sec theta with 1/cos theta in the above expression:

1/(1/cos theta) - tan theta

Simplifying the above expression, we get:

cos theta - sin theta/cos theta

Multiplying both numerator and denominator by cos theta, we get:

(cos theta * cos theta - sin theta)/cos theta

Simplifying further, we get:

cos^2 theta - sin theta/cos theta

Using the identity sin^2 theta + cos^2 theta = 1, we can replace cos^2 theta with 1 - sin^2 theta:

1 - sin^2 theta - sin theta/cos theta

Combining like terms, we get:

(1 - sin theta * sin theta - sin theta)/cos theta

Simplifying further, we get:

(1 - sin theta)(1 + sin theta)/cos theta

Simplifying RHS

sin/cos theta

This can be further simplified as:

sin theta/cos theta

Using the identity tan theta = sin theta/cos theta, we can simplify the above expression as:

tan theta

Proving LHS = RHS

We can now equate the LHS and RHS and show that they are equal:

(1 - sin theta)(1 + sin theta)/cos theta = tan theta

Multiplying both sides by cos theta, we get:

(1 - sin^2 theta) = sin theta

Using the identity sin^2 theta + cos^2 theta = 1, we can replace sin^2 theta with 1 - cos^2 theta:

1 - cos^2 theta = sin theta

Multiplying both sides by -1, we get:

cos^2 theta - 1 = -sin theta

Using the identity cos^2 theta + sin^2 theta = 1, we can replace cos^2 theta with 1 - sin^2 theta:

1 - sin^2 theta - 1 = -sin theta

Simplifying, we get:

-sin^2 theta + sin theta = 0

Factorizing the above expression, we get:

sin theta (1 - sin theta) = 0

This equation is true when sin theta = 0 or sin theta = 1.

When sin theta = 0, the equation reduces to:

cos^2 theta - 1 = 0

Using the identity cos^2 theta + sin^2 theta = 1, we can replace cos^2 theta with 1 - sin^2 theta:

1 - sin^2 theta - 1 = 0

Simplifying, we get:

sin^2 theta = 0

This equation is true only when theta = 0 or theta = pi.

When sin theta = 1, the equation reduces to:

cos^2 theta - 2 = 0

This equation has no real solutions.

Therefore, we can conclude that the given equation is true only when theta = 0 or theta