Without using a calculator, find the exact value of $( ext{sin }22.5^{ iny ext{o}} + ext{cos }22.5^{ iny ext{o}})^2$ You must show each stage of your working - Edexcel - A-Level Maths Pure - Question 28 - 2013 - Paper 1

Starting with the equation $ext{cos } 2 heta + ext{sin } heta = 1$, we can rearrange it:

$ext{cos } 2 heta = 1 - ext{sin } heta$

Using the double angle formula $ext{cos } 2 heta = 2 ext{cos}^2 heta - 1$, we substitute:

$2 ext{cos}^2 heta - 1 = 1 - ext{sin } \theta$

Rearranging leads to:

$2 ext{cos}^2 heta + ext{sin } \theta - 2 = 0$

Using the identity $ext{cos}^2 heta = 1 - ext{sin}^2 heta$ gives:

$2(1 - ext{sin}^2 \theta) + ext{sin } \theta - 2 = 0$

This simplifies to the desired form, where we identify $k = 2$.

From the rearranged equation $k \text{sin}^2 \theta - \text{sin } \theta = 0$ we can factor:

$ext{sin } \theta (k \text{sin } \theta - 1) = 0$

Setting each factor to zero gives:

1. $\text{sin } \theta = 0$, resulting in:

   • $\theta = 0^{\tiny\text{o}}, 180^{\tiny\text{o}}$ (note the restriction $0 < \theta < 360^{\tiny\text{o}}$ excludes 0).

2. $k \text{sin } \theta - 1 = 0$ can be solved to find:

   • $\text{sin } \theta = \frac{1}{2}$. Thus, for $heta$ we have:
   • $\theta = 30^{\tiny\text{o}}$ and $150^{\tiny\text{o}}$.

Summarizing our solutions:

• $\theta = 30^{\tiny\text{o}}, 150^{\tiny\text{o}}, 180^{\tiny\text{o}}$ (all within the required range).