The Pythagorean identity, tan^2(x)+1=sec^2(x), can be used to solve the equation tan(x+1)=sec(x) for 0°≤x<360°.
To solve the equation tan(x+1)=sec(x) for 0°≤x<360°, we can utilize the Pythagorean identity for trigonometric functions, which states that tan^2(x)+1=sec^2(x). Rearranging this identity, we get tan^2(x)=sec^2(x)−1. Substituting this expression into the original equation, we have sec^2 (x)−1+1=sec(x), simplifying to sec^2(x)=sec(x).
Dividing both sides by sec(x), we obtain sec(x)=1. This implies that the angle x has a cosine of 1, which occurs when x=0°. Therefore, the solution to the equation tan(x+1)=sec(x) within the given range is x=0°. This result aligns with the trigonometric properties and the Pythagorean identity, demonstrating how the application of fundamental trigonometric relationships facilitates the solution of the given equation.