To verify the given identity sec(beta) - 1 / cos(beta) = sec(beta), we need to manipulate the left side of the equation to match the right side. By simplifying the expression step by step and using trigonometric identities, we can show that the given equation is true.
To verify the given identity sec(beta) - 1 / cos(beta) = sec(beta), we need to manipulate the left side of the equation to match the right side.
- Start by finding the common denominator of the fractions on the left side, which is cos(beta).
- The expression becomes (sec(beta) - 1)/cos(beta).
- Next, simplify the numerator: sec(beta) - 1 = (1/cos(beta)) - 1 = (1 - cos(beta))/cos(beta).
- Substituting this back into the original expression, we have (1 - cos(beta))/cos(beta) / cos(beta).
- Simplify further by multiplying the numerator and denominator by cos(beta) to get (1 - cos(beta))/(cos^2(beta)).
- Using the identity sec^2(beta) = 1 + tan^2(beta), we rewrite cos^2(beta) = 1 - sin^2(beta) as 1/(1 - sin^2(beta)).
- Therefore, (1 - cos(beta))/(cos^2(beta)) = (1 - cos(beta))/(1 - sin^2(beta)) = sec(beta).
Thus, we have verified that sec(beta) - 1 / cos(beta) = sec(beta).
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