F'<span>(x)</span>=−2<span>cos<span>(x)</span></span><span>sin<span>(x)</span></span>
, <span><span>f''<span>(x)</span>=2<span><span>sin2</span><span>(x)</span></span>−2<span><span>cos2</span><span>(x)</span></span></span>
</span>
<span><span>f'''<span>(x)</span>=4<span>sin<span>(x)</span></span><span>cos<span>(x)</span></span>+4<span>cos<span>(x)</span></span><span>sin<span>(x)</span></span>=8<span>cos<span>(x)</span></span><span>sin<span>(x)</span></span></span>
</span>
<span><span>f''''<span>(x)</span>=−8<span><span>sin2</span><span>(x)</span></span>+8<span><span>cos2</span><span>(x)</span></span></span>
</span>, <span><span>f'''''<span>(x)</span>=⋯=−32<span>cos<span>(x)</span></span><span>sin<span>(x)</span></span></span>
</span>,
<span><span>f''''''<span>(x)</span>=32<span><span>sin2</span><span>(x)</span></span>−32<span><span>cos2</span><span>(x)</span></span></span>
</span>, etc...
Hence, <span><span><span>f<span>(0)</span></span>=1</span>
</span>, <span><span>f'<span>(0)</span>=0</span>
</span>, <span><span>f''<span>(0)</span>=−2</span>
</span>, <span><span>f'''<span>(0)</span>=0</span>
</span>, <span><span>f''''<span>(0)</span>=8</span>
</span>, <span><span>f'''''<span>(0)</span>=0</span>
</span>, <span><span>f''''''<span>(0)</span>=−32</span>
</span>, etc...
Since <span><span>2!=2</span>
</span>, <span><span><span>8<span>4!</span></span>=<span>824</span>=<span>13</span></span>
</span>, and <span><span><span><span>−32</span><span>6!</span></span>=<span><span>−32</span>720</span>=−<span>245</span></span>
</span>, this much calculation leads to an answer of
<span><span>1−<span>x2</span>+<span><span>x4</span>3</span>−<span>245</span><span>x6</span>+⋯</span>
</span>
This does happen to converge for all <span>x
</span> and it does happen to equal <span><span><span>cos2</span><span>(x)</span></span>
</span> for all <span>x
</span>. You can also check on your own that the next non-zero term is <span><span>+<span><span>x8</span>315</span></span>
</span>
2) Use the well-known Maclaurin series <span><span><span>cos<span>(x)</span></span>=1−<span><span>x2</span><span>2!</span></span>+<span><span>x4</span><span>4!</span></span>−<span><span>x6</span><span>6!</span></span>+⋯</span>
</span>
<span><span>=1−<span><span>x2</span>2</span>+<span><span>x4</span>24</span>−<span><span>x6</span>720</span>+⋯</span>
</span> and multiply it by itself (square it).
To do this, first multiply the first term <span>1
</span> by everything in the series to get
<span><span><span><span>cos2</span><span>(x)</span></span>=<span>(1−<span><span>x2</span>2</span>+<span><span>x4</span>24</span>−<span><span>x6</span>720</span>+⋯)</span>+⋯</span>
</span>
Next, multiply <span><span>−<span><span>x2</span>2</span></span>
</span> by everything in the series to get
<span><span><span><span><span>cos2</span><span>(x)</span></span>=<span>(1−<span><span>x2</span>2</span>+<span><span>x4</span>24</span>−<span><span>x6</span>720</span>+⋯)</span>+<span>(−<span><span>x2</span>2</span>+<span><span>x4</span>4</span>−<span><span>x6</span>48</span>+<span><span>x8</span>1440</span>+</span></span><span><span>⋯)</span>+⋯</span></span>
</span>
Then multiply <span><span><span>x4</span>24</span>
</span> by everything in the series to get
<span><span><span><span><span>cos2</span><span>(x)</span></span>=<span>(1−<span><span>x2</span>2</span>+<span><span>x4</span>24</span>−<span><span>x6</span>720</span>+⋯)</span>+<span>(−<span><span>x2</span>2</span>+<span><span>x4</span>4</span>−<span><span>x6</span>48</span>+<span><span>x8</span>1440</span>+</span></span><span><span>⋯)</span>+<span>(<span><span>x4</span>24</span>−<span><span>x6</span>48</span>+<span><span>x8</span>576</span>−<span><span>x10</span>17280</span>+⋯)</span>+⋯</span></span>
</span>
etc...
If you go out far enough and combine "like-terms", using the facts, for instance, that <span><span>−<span>12</span>−<span>12</span>=−1</span>
</span> and <span><span><span>124</span>+<span>14</span>+<span>124</span>=<span>824</span>=<span>13</span></span>
</span>, etc..., you'll eventually come to the same answer as above:
<span>1−<span>x2</span>+<span><span>x4</span>3</span>−<span>245</span><span>x6</span>+<span>⋯</span></span>
