Verify the identity shown below
1/(sec(theta) - tan theta) = sec(theta) + tan theta
clear the fraction, and recall that
sec^2 - tan^2 = 1
To verify this identity, we need to simplify the left-hand side (LHS) and the right-hand side (RHS) of the equation separately and then show that they are equal.
Starting with the LHS:
1/(sec(theta) - tan(theta))
We can use a common trigonometric identity to simplify this expression. The identity states that:
sec(theta) = 1/cos(theta)
Using this identity, we can rewrite the LHS as:
1/(1/cos(theta) - tan(theta))
To simplify further, we need to find a common denominator for the fractions. The common denominator is cos(theta), so we multiply the numerator and denominator of the second fraction by cos(theta):
1/(1/cos(theta) - tan(theta)) * (cos(theta)/cos(theta))
Simplifying the expression by multiplying the numerators and denominators gives:
cos(theta)/[1 - cos(theta) * tan(theta)]
Now, we need to simplify the RHS:
sec(theta) + tan(theta)
Using the identity sec(theta) = 1/cos(theta), we can rewrite this as:
1/cos(theta) + sin(theta)/cos(theta)
Now, we have a common denominator for the fractions, so we can combine them:
(1 + sin(theta))/cos(theta)
Therefore, the simplified LHS is:
cos(theta)/[1 - cos(theta) * tan(theta)]
And the simplified RHS is:
(1 + sin(theta))/cos(theta)
To complete the verification, we can show that the LHS is equal to the RHS by demonstrating that their expressions are equivalent:
cos(theta)/[1 - cos(theta) * tan(theta)] = (1 + sin(theta))/cos(theta)
To do this, we can multiply both sides of the equation by cos(theta) to eliminate the denominators:
cos(theta) * [cos(theta)/[1 - cos(theta) * tan(theta)]] = cos(theta) * [(1 + sin(theta))/cos(theta)]
Simplifying both sides gives:
cos(theta) = 1 + sin(theta)
Which is true based on one of the fundamental trigonometric identities. Therefore, we have verified the given identity.