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# PERSONAL USE

1. $\cot ^{ 2 }{ \alpha } \tan ^{ 2 }{ \alpha } +\cot ^{ 2 }{ a }$ $=\cot ^{ 2 }{ \alpha } (\tan ^{ 2 }{ \alpha } +1)$ $=\cot ^{ 2 }{ \alpha } (\sec ^{ 2 }{ \alpha } )$ $=\frac { 1 }{ \sin ^{ 2 }{ \alpha } } =\csc ^{ 2 }{ \alpha }$ 2. $\sec \phi (\frac { \cos { \phi } }{ \tan { \phi } } )$ $=\frac { 1 }{ \cos { \phi } } (\cos { \phi } \cdot \frac { \cos { \phi } }{ \sin { \phi } } )$ $=\frac { \cos { \phi } }{ \sin { \phi } } =\cot { \phi }$ 3. $x^2-25=(5\csc\theta)^2-25$ $=25\csc^2\theta-25=25(\csc^2\theta-1)$ $=25(\cot^2\theta)$ $\therefore \sqrt{25\cot^2\theta}=5\cot\theta$ 4. $\cos (\frac { \pi }{ 2 } -x)\csc ^{ 2 } x=\sin { x } (\frac { 1 }{ \sin ^{ 2 }{ x } } )$ $=\frac { 1 }{ \sin { x } } =\csc { x }$ 5. $\sin^2\alpha\csc^2\alpha-\sin^2\alpha=\sin^2\alpha(\csc^2\alpha-1)$ $=\sin ^{ 2 } \alpha (\cot ^{ 2 } \alpha )=\sin ^{ 2 } \alpha (\frac { \cos ^{ 2 }{ \alpha } }{ \sin ^{ 2 }{ \alpha } } )$ $=\cos^2\alpha$ 6. $\cot^2\alpha\cos^2\alpha-\cot^2\alpha=0$ $\cot^2\alpha(\cos^2\alpha-1)=0$ $\cot^2\alpha(-\sin^2\alpha)=0$ $(\frac { \cos ^{ 2 }{ \alpha } }{ \sin ^{ 2 }{ \alpha } } )(-\sin^2\alpha)=0$ $-\cos^2\alpha=0$ $\alpha =\frac { \pi }{ 2 } ,-\frac { \pi }{ 2 }$ 7. $\sin^3\alpha(\csc\alpha+\cot^3\alpha\sec\alpha)$ $=\sin^3\alpha(\frac{1}{\sin\alpha}+\cot^2\alpha(\cot\alpha)(\frac{1}{\cos\alpha})$ $=\sin^3\alpha(\frac{1}{\sin\alpha}+\frac{\cos^2\alpha}{\sin^3\alpha})$ $=\sin^2\alpha+\cos^2\alpha=1$ 8. $3(\cos1-\sin^2x)(\tan^2x+1)=3(\cos^2x)(\frac{1}{\cos^2x})=3$ 10. $\frac{2}{\cot{x}-\csc{x}}=\frac{2(\cot{x}+\csc{x})}{\cot^2{x} -\csc^2{x}}=-2\cot{x}-2\csc{x}$ 11. Similar to 5

12. Similar to 3

13. Split the Equation up. Then use the unit circle to simplify.

14. $\ln { \left| \sin \theta \right| } +\ln { \left| \sec \theta \right| } =\ln { \left| \sin \theta \cdot \sec \theta \right| } =\ln { \left| \tan \theta \right| }$ 15. Should be able to do it. Split the numerator up.

16. Should be able to do it.

17. Right idea you have there. Just make one over tangent into cosine over sine. Then simplify.

18. Should be able to do it.

19. Use product to sum formulas.

20. You should do it. Hint: Convert tangent squared into one minus secant squared.

Note by Yagna Patel
1 year, 10 months ago