506 CHAPTER 7 Analytic Trigonometry Establishing an Identity Establish the identity: csc tan sec θ θ θ ⋅ = Solution EXAMPLE 2 Start with the left side, because it contains the more complicated expression. Then use a reciprocal identity and a quotient identity. csc tan 1 sin sin cos 1 cos sec θ θ θ θ θ θ θ ⋅ = ⋅ = = The right side has been reached, so the identity is established. Establishing an Identity Establish the identity: sin cos 1 2 2 θ θ ( ) ( ) − + − = Solution EXAMPLE 3 Begin with the left side and, because the arguments are ,θ− use Even–Odd Identities. sin cos sin cos sin cos sin cos 1 2 2 2 2 2 2 2 2 θ θ θ θ θ θ θ θ ( ) ( ) [ ( )] [ ( )] ( ) ( ) ( ) ( ) − + − = − + − = − + = + = Even–Odd Identities Pythagorean Identity Establishing an Identity Establish the identity: sin cos sin cos cos sin 2 2 θ θ θ θ θ θ ( ) ( ) ( ) ( ) − − − − − − = − Solution EXAMPLE 4 The left side contains the more complicated expression. Also, the left side contains expressions with the argument ,θ− whereas the right side contains expressions with the argument .θ So start with the left side and use Even–Odd Identities. sin cos sin cos sin cos sin cos sin cos sin cos sin cos sin cos sin cos sin cos sin cos cos sin 2 2 2 2 2 2 2 2 θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ ( ) ( ) ( ) ( ) [ ( )] [ ( )] ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) − − − − − − = − − − − − − = − − − − = − − − = − + − + = − Even—Odd Identities Simplify. Factor. Divide out and simplify. Establishing an Identity Establish the identity: u u u 1 tan 1 cot tan + + = Solution EXAMPLE 5 u u u u u u u u u u u 1 tan 1 cot 1 tan 1 1 tan 1 tan tan 1 tan tan 1 tan tan 1 tan ( ) + + = + + = + + = + + = Now Work PROBLEM 21 Now Work PROBLEMS 25 AND 29
RkJQdWJsaXNoZXIy NjM5ODQ=