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9. Find: \(sin{\frac{x}{2}}, cos{\frac{x}{2}}\) and \(tan{\frac{x}{2}}\), if \(cos{x}=-\frac{1}{3}\), x in quadrant III.

Trigonometric Functions

Miscellaneous Exercise

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Class 11

Trigonometric Functions

Miscellaneous Exercise

1. Prove that: \(2cos{\frac{\pi}{13}}cos{\frac{9\pi}{13}}+cos{\frac{3\pi}{13}}+cos{\frac{5\pi}{13}}=0\).

2. Prove that: \(\left(\sin{3x}+\sin{x}\right)\sin{x}+\left(\cos{3x}-\cos{x}\right)\cos{x}=0\).

3. Prove that: \(\left(cos{x}+cos{y}\right)^2+\left(sin{x}-sin{y}\right)^2=4{cos}^2{\frac{x+y}{2}}\).

4. Prove that: \(\left(cos{x}-cos{y}\right)^2+\left(sin{x}-sin{y}\right)^2=4{sin}^2{\frac{x-y}{2}}\).

5. Prove that: \(sin{x}+sin{3x}+sin{5x}+sin{7x}=4cos{x} cos{2x} sin{4x}\).

6. Prove that: \(\frac{\left(sin{7}x+sin{5}x\right)+\left(sin{9}x+sin{3}x\right)}{\left(cos{7}x+cos{5}x\right)+\left(cos{9}x+cos{3}x\right)}=tan{6}x\).

7. Prove that: \(sin{3}x+sin{2}x-sin{x}=4sin{x}cos{\frac{x}{2}}cos{\frac{3x}{2}}\).

8. Find: \(sin{\frac{x}{2}}, cos{\frac{x}{2}}\) and \(tan{\frac{x}{2}}\), if \(tan{x}=-\frac{4}{3}\), x in quadrant II.

9. Find: \(sin{\frac{x}{2}}, cos{\frac{x}{2}}\) and \(tan{\frac{x}{2}}\), if \(cos{x}=-\frac{1}{3}\), x in quadrant III.

10. Find: \(sin{\frac{x}{2}}, cos{\frac{x}{2}}\) and \(tan{\frac{x}{2}}\), if \(sin\ x{=}\frac{1}{4}\), x in quadrant II.