In trigonometry, it is really important to know and able to use identities. One of the main and crucial categories of identities is **Double Angle** **Formula**. In this post, we talked about different trig identities and formulas. **Double Angle** **Formulas** are trigonometric identities that simplify a trigonometric function of $2x$ as of trigonometric functions of $x$. First we recommend to visit the previous post to review identities. Here in this post, first we recall the **Double Angle** **Formula** and then we see some examples.

**Double Angle Formulas:**

Here we have main **Double Angle** **Formulas**.

- $\sin (2 u) =2 \sin u \cos u$.
- $\cos (2 u) =\cos ^{2} u-\sin ^{2} u$.
- $\cos (2 u) =2 \cos ^{2} u-1$.
- $\cos (2 u) =1-2 \sin ^{2} u$.
- $\tan (2 u) =\dfrac{2 \tan u}{1-\tan ^{2} u}$.

**Examples of Double Angle Formulas**

Before checking the solutions, try to solve it by yourself first.

**Example 1:** If $\sin x=\dfrac{3}{5}$ and $x$ is in the first quadrant, find $\tan (2 x)$.

**Solution:** Since $x$ is in the first quadrant, both $\sin x$ and $\cos x$ are positive. Now we use the **Pythagorean identity **$\sin^{2} x+\cos^{2} x=1$ to find $\cos x$. From this identity we have, $\dfrac{9}{25} +\cos^{2} x=1$, so $\cos x=\dfrac{4}{5}$. Then $\tan x=\dfrac{\sin x}{\cos x}=\dfrac{\dfrac{3}{5}}{\dfrac{4}{5}}=\dfrac{3}{4}$. Now by the last **Double Angle Formula**, we have $\tan (2 u)=\dfrac{2 \dfrac{3}{4}}{1-\dfrac{9}{16} }=\dfrac{\dfrac{3}{2}}{\dfrac{7}{16}}=\dfrac{48}{14}=\dfrac{24}{7}$.

**Example 2:** Prove the following identity $\dfrac{1-\cos 2x}{1+\cos 2x}=\tan^2 x$.

**Solution:** To solve this problem we use the double angle formula for $\cos 2x$. We simplify the left side to get the right side. By the double angle formula we have $\dfrac{1-\cos 2x}{1+\cos 2x}=\dfrac{1-(1-2 \sin ^{2} x)}{1+2 \cos ^{2} x-1}=\dfrac{\sin ^{2} x}{\cos ^{2} x}=\tan^2 x$.

Here we add more examples without solutions for you to practice.

**Example 3:** Prove the identity: $\dfrac{\cos 2x}{\cos x- \sin x}=\cos x+\sin x$.

The following example is really common in exams.

**Example 4:** Solve $ \cos(2x)=\cos(x)$ when $0\leq x <2\pi$.

The following example is a bit challenging, but give it a try.

**Example 5:** Prove the identity: $\cos^4{x}=\dfrac{\cos(4x)}{8}+\dfrac{\cos(2x)}{2}+\dfrac{3}{8}$.

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