Solutions for Sum and Difference Identities
Solutions to Try Its
1. [latex]\frac{\sqrt{2}+\sqrt{6}}{4}[/latex] 2. [latex]\frac{\sqrt{2}-\sqrt{6}}{4}[/latex] 3. [latex]\frac{1-\sqrt{3}}{1+\sqrt{3}}[/latex] 4. [latex]\cos \left(\frac{5\pi }{14}\right)[/latex] 5. [latex]\begin{array}{l}\tan \left(\pi -\theta \right)=\frac{\tan \left(\pi \right)-\tan \theta }{1+\tan \left(\pi \right)\tan \theta }\hfill \\ \text{ }=\frac{0-\tan \theta }{1+0\cdot \tan \theta }\hfill \\ \text{ }=-\tan \theta \hfill \end{array}[/latex]Solutions to Odd-Numbered Answers
1. The cofunction identities apply to complementary angles. Viewing the two acute angles of a right triangle, if one of those angles measures [latex]x[/latex], the second angle measures [latex]\frac{\pi }{2}-x[/latex]. Then [latex]\sin x=\cos \left(\frac{\pi }{2}-x\right)[/latex]. The same holds for the other cofunction identities. The key is that the angles are complementary. 3. [latex]\sin \left(-x\right)=-\sin x[/latex], so [latex]\sin x[/latex] is odd. [latex]\cos \left(-x\right)=\cos \left(0-x\right)=\cos x[/latex], so [latex]\cos x[/latex] is even. 5. [latex]\frac{\sqrt{2}+\sqrt{6}}{4}[/latex] 7. [latex]\frac{\sqrt{6}-\sqrt{2}}{4}[/latex] 9. [latex]-2-\sqrt{3}[/latex] 11. [latex]-\frac{\sqrt{2}}{2}\sin x-\frac{\sqrt{2}}{2}\cos x[/latex] 13. [latex]-\frac{1}{2}\cos x-\frac{\sqrt{3}}{2}\sin x[/latex] 15. [latex]\csc \theta [/latex] 17. [latex]\cot x[/latex] 19. [latex]\tan \left(\frac{x}{10}\right)[/latex] 21. [latex]\sin \left(a-b\right)=\left(\frac{4}{5}\right)\left(\frac{1}{3}\right)-\left(\frac{3}{5}\right)\left(\frac{2\sqrt{2}}{3}\right)=\frac{4 - 6\sqrt{2}}{15}[/latex] [latex-display]\cos \left(a+b\right)=\left(\frac{3}{5}\right)\left(\frac{1}{3}\right)-\left(\frac{4}{5}\right)\left(\frac{2\sqrt{2}}{3}\right)=\frac{3 - 8\sqrt{2}}{15}[/latex-display] 23. [latex]\frac{\sqrt{2}-\sqrt{6}}{4}[/latex] 25. [latex]\sin x[/latex]![Graph of y=sin(x) from -2pi to 2pi.](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/923/2015/04/25001510/CNX_Precalc_Figure_07_02_201.jpg)
![Graph of y=cot(pi/6 - x) from -2pi to pi - in comparison to the usual y=cot(x) graph, this one is reflected across the x-axis and shifted by pi/6.](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/923/2015/04/25001512/CNX_Precalc_Figure_07_02_203.jpg)
![Graph of y=cot(pi/4 + x) - in comparison to the usual y=cot(x) graph, this one is shifted by pi/4.](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/923/2015/04/25001513/CNX_Precalc_Figure_07_02_205.jpg)
![Graph of y = sin(x) / rad2 + cos(x) / rad2 - it looks like the sin curve shifted by pi/4.](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/923/2015/04/25001515/CNX_Precalc_Figure_07_02_207.jpg)
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- Precalculus. Provided by: OpenStax Authored by: OpenStax College. Located at: https://cnx.org/contents/[email protected]:1/Preface. License: CC BY: Attribution.