Show that the following are not trigonometric identities
1.tan 2x = 2tan x
2. sec x= sqr rt 1+tan^2 x
3. sin(x+y)=sin x +sin y
pick an angle and show that the equation fails.
tan(pi/2) ≠ 2 tan(pi/4)
To show that the given equations are not trigonometric identities, we need to find at least one counterexample where the equation does not hold true for all values of the variables involved.
1. tan 2x = 2tan x:
To prove that this equation is not a trigonometric identity, we can find a value of x for which the equation does not hold true. Let's assume x = 45° (or π/4 radians):
tan(2 * 45°) = tan(90°) = ∞ (Since the tangent of 90° is undefined)
2 * tan(45°) = 2 * 1 = 2
As we can see, tan(2x) ≠ 2tan(x) when x = 45°, so this equation is not an identity.
2. sec x = √(1 + tan^2 x):
Similarly, let's find a counterexample for this equation. Assume x = 0° (or 0 radians):
sec(0°) = 1
√(1 + tan^2(0°)) = √(1 + 0) = √1 = 1
In this case, sec(x) = √(1 + tan^2(x)) holds true when x = 0°, so this equation is an identity.
3. sin(x + y) = sin(x) + sin(y):
To show that this equation is not a trigonometric identity, we need to find values of x and y for which the equation does not hold true. Assume x = 0° (or 0 radians) and y = 90° (or π/2 radians):
sin(0° + 90°) = sin(90°) = 1
sin(0°) + sin(90°) = 0 + 1 = 1
In this case, sin(x + y) = sin(x) + sin(y) holds true when x = 0° and y = 90°, so this equation is an identity.
Therefore, out of the three given equations, only the first one, tan(2x) = 2tan(x), is not a trigonometric identity.