Exercise 21 Page 883

We want to verify the given trigonometric identity. sec θ csc θ = tan θ + cot θ We will simplify the left-hand side of the equation using the Reciprocal Identities. Then we will simplify the right-hand side of the equation using the Quotient and the Pythagorean Identities to arrive at the left-hand side. Let's start by recalling two of the Reciprocal Identities. sec θ = 1/cos θ, cos θ ≠ 0 [1.0 em] csc θ = 1/sin θ, sin θ ≠ 0We can substitute 1cos θ for sec θ and 1sin θ for csc θ in our expression on the left-hand side. Then we will simplify it.

Now we can move on to simplifying the expression on the right-hand side. Let's recall the Quotient Identities. tan θ = sin θ/cos θ, cos θ ≠ 0 [1.0 em] cot θ = cos θ/sin θ, sin θ ≠ 0 We can substitute sin θcos θ for tan θ and cos θsin θ for cot θ in our expression. Then we will simplify it.

Finally, let's recall one of the Pythagorean Identities. cos^2 θ + sin ^2 θ = 1 We will substitute 1 for sin ^2 θ + cos^2 θ in our expression.

We obtained the expression on the right-hand side that is equal to simplified form of the expression on the left-hand side. Therefore, we have verified the identity.