Final answer:
To demonstrate that secθ+tanθ=1/x, we first write secθ and tanθ in terms of cosθ and sinθ and use the given equation secθ−tanθ=x. After rearranging and simplifying, we solve for cosθ and sinθ using trigonometric identities and the Pythagorean identity.
Step-by-step explanation:
If secθ−tanθ=x, we want to show that secθ+tanθ=1/x. To do this, we will use the trigonometric identity secθ = 1/cosθ and tanθ = sinθ/cosθ. Rearranging the given equation, we get secθ = x + tanθ. Substituting the trigonometric identities, we have 1/cosθ = x + sinθ/cosθ. Simplifying, we find 1/cosθ = (x∙cosθ + sinθ)/cosθ, which leads us to 1 = x∙cosθ + sinθ. Solving for cosθ, we get cosθ = (1 - sinθ)/x. Squaring both sides, we get cos²θ = (1 - 2sinθ + sin²θ)/x², and since we know that cos²θ + sin²θ = 1 (Pythagorean identity), we can substitute for cos²θ.
Thus, 1 = (1 - 2sinθ + sin²θ)/x² + sin²θ, resulting in x² = 1 - 2sinθ + 2sin²θ. From the initial identity, we also know that tanθ = sinθ/cosθ = sinθ/((1 - sinθ)/x) = x∙sinθ/(1 - sinθ). Then secθ + tanθ = (1/cosθ) + (sinθ/cosθ), which can be written as 1/cosθ + sinθ/cosθ = (1 + sinθ)/cosθ. Since we have secθ−tanθ=x, we multiply this by its reciprocal to get secθ + tanθ as 1/x. Also, with the value of cosθ we found earlier, we have cosθ = (1 - sinθ)/x and sinθ can be found using the equation derived from squaring cosθ and solving for sinθ.