In trigonometry, several identities connect the reciprocal and quotient functions to sinθ and cosθ.
Answer the following questions about one such identity.
(1) Prove algebraically that sec2θ−1=tan2θ for every value of θ for which the expressions are defined.
(2) Near θ=0, both sec2θ−1 and tan2θ approach 0. Explain why their rates of approach are comparable and discuss how the equality from part (a) influences the evaluation of limits that initially present the indeterminate form 0/0. Support your explanation with clear reasoning grounded in the continuity of the trigonometric functions involved.