What trigonometric identities was used to get this? [closed]

Question

I have this solution for a math question, but I didn't get how did we get from the line $3$ to $4$ and also from $4$ to $5$. I can understand that this is done through trigonometric identities. but which ones?

The question is

Answer 1

The entire calculation seems highly suspect to me, not just due to the (not entirely consistent) use of trig identities, but also due to an earlier step.

I will use $i$ for the imaginary unit. I will also insert some parentheses since I really dislike reading notation like $i\pi/2t$.

In the absence of a definition of $X$ I will have to assume that the derivation of this equation from the summation form is correct: $$ x(t) = 6 + (-4 - 2i) e^{i(\pi/2)t} + 2 e^{i(2\pi/2)t} + (-4 + 2i) e^{i(3\pi/2)t}. \tag1 $$ On the next line it is claimed that $$ x(t) \stackrel?= 6 + (-4 - 2i) e^{i(\pi/2)t} + (-4 + 2i) e^{-i(\pi/2)t} + 2 e^{i\pi t}. \tag2 $$ Observing that the first, second, and third terms on the right side of Equation $(1)$ are equal to the first, second, and fourth terms on the right side of Equation $(2)$, the claim in Equation $(2)$ comes down to a claim that $$ (-4 + 2i) e^{i(3\pi/2)t} \stackrel?= (-4 + 2i) e^{-i(\pi/2)t}. \tag{*} $$ Since $3\pi/2 = -\pi/2 + 2\pi,$ Equation $(\text{*})$ is true for every integer value of $t$, but it is false for every other value of $t$.

The next equation is a simple rearrangement of terms from Equation $(2)$: $$ x(t) \stackrel?= 6 - 4\left(e^{i(\pi/2)t} + e^{-i(\pi/2)t}\right) - 2i \left(e^{i(\pi/2)t} - e^{-i(\pi/2)t}\right) + 2 e^{i\pi t}. \tag3 $$

From this we are supposed to conclude that $$ x(t) \stackrel?= 6 - 8 \cos((\pi/2)t) + 4 \sin((\pi/2)t) + 2\cos(\pi t). \tag4 $$ The first, second, and third terms of Equation $(4)$ are equal to the first three terms of Equation $(3)$ by well-known trig identities. In order to derive Equation $(4)$ from Equation $(3)$, then, we must believe that $$ 2 e^{i\pi t} \stackrel?= 2\cos(\pi t). \tag{**} $$ Since $\cos(\theta) = \Re\left(e^{i\theta}\right),$ the real parts of both sides of Equation $(\text{**})$ match up all right, but the imaginary parts do not. Once again we have an equation that is true for every integer $t$ but false for every other value of $t$.

If the final answer is correct it seems to me it could only be through previous lucky errors in the calculations before the final sequence of equations in the question.

Answer 2

The identities are...

• $e^{jt} = \cos t + j \sin t$
• $\Re (e^{jt}) = \cos jt = \dfrac {e^{jt} + e^{-jt}}{2}$ and $\Im (e^{jt}) = \sin jt = \dfrac {e^{jt} - e^{-jt}}{2i}$
• $a \sin t + b \cos t = \sqrt {a^2 + b^2} \cos\left(t + \arctan \left(-\dfrac {b}{a}\right)\right)$

Note: I am using $j$ here as in the exercise, but usually the imaginary unit for $\sqrt {-1}$ is $i$.