Verify the identity:
tanx/secx-1=secx+1/tanx
To verify the identity tanx/secx-1 = secx+1/tanx, we need to manipulate the expression on one side of the equation to match the expression on the other side.
Let's start by working on the left side of the equation: tanx/secx-1.
1. Recall the trigonometric identity: tan(x) = sin(x)/cos(x). We can substitute this into our equation:
(sinx/cosx) / secx - 1
2. Recall the trigonometric identity: sec(x) = 1/cos(x). We can substitute this into our equation:
(sinx/cosx) / (1/cosx) - 1
3. Simplify the expression:
sinx / 1 - cosx / cosx
4. Combine the terms with a common denominator:
(sinx - cosx) / cosx
Now let's work on the right side of the equation: secx+1/tanx.
1. Recall the trigonometric identity: sec(x) = 1/cos(x). We can substitute this into our equation:
1/cosx + 1/(sinx/cosx)
2. Simplify the expression:
1/cosx + cosx/sinx
3. Combine the terms with a common denominator:
(sinx + cosx) / (cosx*sinx)
Now, we can see that the expression on the left side is equal to the expression on the right side:
(sin(x) - cos(x)) / cos(x) = (sin(x) + cos(x)) / (cos(x) * sin(x))
Therefore, we have verified the identity.