Final answer to the problem
Step-by-step Solution
How should I solve this problem?
- Express everything into Sine and Cosine
- Prove from LHS (left-hand side)
- Prove from RHS (right-hand side)
- Exact Differential Equation
- Linear Differential Equation
- Separable Differential Equation
- Homogeneous Differential Equation
- Integrate by partial fractions
- Product of Binomials with Common Term
- FOIL Method
- Load more...
I. Express the LHS in terms of sine and cosine and simplify
Start from the LHS (left-hand side)
Using the sine double-angle identity: $\sin\left(2\theta\right)=2\sin\left(\theta\right)\cos\left(\theta\right)$, where $2\theta=4$
II. Express the RHS in terms of sine and cosine and simplify
Start from the RHS (right-hand side)
Simplify $2\sin\left(2\right)\cos\left(2\right)$ using the trigonometric identity: $\sin(2x)=2\sin(x)\cos(x)$
Multiply $2$ times $2$
Simplify the fraction $\frac{2\sin\left(4\right)}{2}$ by $2$
III. Choose what side of the identity are we going to work on
To prove an identity, we usually begin to work on the side of the equality that seems to be more complicated, or the side that is not expressed in terms of sine and cosine. In this problem, we will choose to work on the left side $2\sin\left(2\right)\cos\left(2\right)$ to reach the right side $\sin\left(4\right)$
Simplify $2\sin\left(2\right)\cos\left(2\right)$ using the trigonometric identity: $\sin(2x)=2\sin(x)\cos(x)$
Multiply $2$ times $2$
Simplify the fraction $\frac{2\sin\left(4\right)}{2}$ by $2$
IV. Check if we arrived at the expression we wanted to prove
Since we have reached the expression of our goal, we have proven the identity