### Power Reduction Calculator

### Power Reduction Identity Tutorial

**The Power Reduction formulas are shown below:**

\text{sin}^2\text{(u)}=\frac{1-(1-2\cdot \text{sin}^2\text{(u)})}{2}

\text{cos}^2\text{(u)}=\frac{1+(1-2\text{sin}^2(\text{u}))}{2}

\tan ^2\text{(u)}=\frac{1+(1-2\text{sin}^2(\text{u}))}{1-(1-2\text{sin}^2(\text{u}))}

\csc ^2\text{(u)}=\frac{2}{1-(1-2\text{sin}^2(\text{u}))}

\sec ^2\text{(u)}=\frac{2}{1+(1-2\text{sin}^2(\text{u}))}

\text{cot}^2\text{(u)}=\frac{1-(1-2\text{sin}^2(\text{u}))}{1+(1-2\text{sin}^2(\text{u}))}

To find the exact value of any of the power reduction identites given a ratio given a ratio and quadrant can be done. For example, to find the exact value of $sin^2u$ given tanu = $\frac{5}{2}$ in quadrant 3 you first have to find the values required for the identity. According to the formula for the power reduction formula for sin, the exact value of sin is needed. This means that the first step in solving the identity is finding the exact value of sin in quadrant 3. Since the quadrant of the given function is 3, the exact value of sin must be negative. The calculation process for finding sin in quadrant 1 is shown below:

\text{tan}=\frac{5}{2}=\frac{\text{opposite}}{\text{adjacent}}

\text{ opposite}=5

\text{hypotenuse}=\sqrt{\text{opposite}^2+\text{adjacent}^2}

\text{hypotenuse}=\sqrt{5^2+2^2}

\text{hypotenuse}=\sqrt{29}

\text{sin}=\frac{\text{opposite}}{\text{hypotenuse}}=-\frac{5\sqrt{29}}{29}

\text{sin}=-\frac{5\sqrt{29}}{29}

\text{(sin must be negative in quadrants 3 and 4)}

\text{(in simplest form)}

Now that the exact value of sin in quadrant three is known, it can be substituted into the formula for $sin^2u$. The step by step process of this is shown below:

\text{sin}^2\text{u}=\frac{1-(1-2\cdot \text{sin}^2\text{u})}{2}

\text{sin}^2\text{u}=\frac{1-(1-2(-\frac{5\sqrt{29}}{29})^2)}{2}

\text{sin}^2\text{u}=\frac{25}{29}

\text{(in simplest form)}

**Tutorial two**

To solve power reduction identites without given values and getting decimal approximations is an easier process. To find these values you simply square any of the trigonometric function values. The value of $\text{sin}^2\13$ is the same as $\sin(13)\cdot\sin(13)$. If told to find the value of $sin^2(22)$ the calculation process below can be used. The value of $\text{cos}^2\text{(u)}$ given cscu = $\frac{3}{2}$ being that the angle is in quadrant 2, the first step would be to know the formula. The formula for $\text{cos}^2\text{(u)}$ is $\frac{1+(1-2\text{sin}^2(\text{u}))}{2}$. According to this formula the only value needed for the calculation is $\sin$. To find $\sin$ in quadrant 2 you must first find the opposite side length since that is needed for the value of sin. to find the opposite side length the calculation process for finding the opposite side length is shown below:

\text{csc}=\frac{3}{2}=\frac{\text{hypotenuse}}{\text{opposite}}

\text{ hypotenuse}=3

\text{sin}=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{2}{3}

\text{sin}=\frac{2}{3}

Now that the opposite side length is known, the value of sin can be calculated. since sin is the ratio of the opposite side length and the hypotenuse, the opposite side length will be the numerator and the hypotenuse will be the denominator. The final part of solving this identity is substituting the value of sin into the equation. The calculation process for this is shown below:

\text{sin}^2\text{u}=\frac{1-(1-2\cdot \text{sin}^2\text{u})}{2}

\text{sin}^2\text{u}=\frac{1-(1-2(\frac{2}{3})^2)}{2}

\text{sin}^2\text{u}=\frac{4}{9}