How to verify this identity:
1+ tan x
(over) = secant x
sine x + cos x
To verify the given identity:
1 + tan x
-----------
sine x + cos x
We can start with the left-hand side (LHS) and try to simplify it to match the right-hand side (RHS). Here's how you can do it:
1. Start with the LHS:
LHS = 1 + tan(x) / (sin(x) + cos(x))
2. Multiply the numerator and denominator of the fraction by the conjugate of (sin(x) + cos(x)):
LHS = (1 + tan(x)) * (sin(x) - cos(x)) / [(sin(x) + cos(x)) * (sin(x) - cos(x))]
3. Use the identity sin^2(x) - cos^2(x) = 1 to simplify the numerator:
LHS = (1 + tan(x)) * (sin(x) - cos(x)) / (sin^2(x) - cos^2(x))
4. Expand and simplify the numerator:
LHS = sin(x) - cos(x) + tan(x) * sin(x) - tan(x) * cos(x) / (sin^2(x) - cos^2(x))
5. Rewrite the denominator using the difference of squares identity:
LHS = sin(x) - cos(x) + tan(x) * sin(x) - tan(x) * cos(x) / [(sin(x) + cos(x)) * (sin(x) - cos(x))]
6. Cancel out the common terms in the numerator and denominator:
LHS = [sin(x) * (1 + tan(x))] / (sin(x) + cos(x))
7. Apply the identity sec(x) = 1 / cos(x) to simplify the numerator:
LHS = [sec(x) * cos(x)] / (sin(x) + cos(x))
8. Simplify further by multiplying the numerator and denominator by sec(x):
LHS = sec(x) * cos(x) / (sec(x) * (sin(x) + cos(x)))
9. Cancel out the common terms in the numerator and denominator:
LHS = cos(x) / (sin(x) + cos(x))
Finally, we can see that the simplified LHS is equal to the RHS, which is sec(x). Therefore, the identity is verified.
Note: In this process, it's important to be familiar with trigonometric identities and algebraic manipulations to simplify and transform the given expression.