Trigonometric Problems

Exercise 1: Prove that:
A) $\beta - \alpha = \frac{\pi}{4}$ , if $sin \alpha = \frac{\sqrt{5}}{5}$ , if α I quadrant, $cos \beta = \frac{\sqrt{10}}{10}$ , β I quadrant.
B) $\alpha + \beta = \frac{\pi}{2}$ , if $sin \alpha = \frac{15}{17}$ , if α I quadrant, $cos \beta = \frac{8}{17}$ , β I quadrant.
C) $\alpha + \beta + \gamma = \frac{\pi}{2}$ , if $sin \alpha = \frac{1}{3}$ , $sin \beta = \frac{\sqrt{11}}{33}$ and $sin \gamma = \frac{3\sqrt{11}}{11}$ , where α, β, γ I quadrant.

Exercise 2: Prove that if $\alpha + \beta + \gamma = \frac{\pi}{2}$ , than
A) tgα.tgβ + tgβ.tgγ + tgα.tgγ = 1.
B) ctgα + ctgβ + ctgβ = ctgα.ctgβ.ctgγ.

Exercise 3: Prove that if
$\alpha + \beta = \frac{\pi}{4}$
then (1 + tgα)(1 + tgβ) = 2.

Exercise 4: Prove that
A) cosα + √3sinα ≤ 2.
B) √2(sinα + cosα) ≤ 2.
C) sin2αcosβ + cos2αsinβ ≤ 1.
D) sin(α + β) ≤ sinα + sinβ.
E) cos(α + β) < cosα - sinα.sinβ.
F) √3cos&lpha; - sinα ≤ 2.
G) sinα - cosα ≤ √2.

Exercise 5: Find the maximal value of y = √3sin2x - cos2x and the value of x, if 0 ≤ x ≤ π.

Exercise 6: Calculate the examples
A) sin67°.cos68° - cos67°.sin68°.
B) sin27°.cos33° + sin63°.cos57° + cos30°.
C) $\frac{tg66^\circ . ctg36^\circ - 1}{tg66^\circ + ctg36^\circ }$
D) $\frac{tg92^\circ + ctg2^\circ }{1 - tg92^\circ . ctg2^\circ}$
E) cos1° + cos121° + cos241°.
F) $\frac{sin110^\circ .sin250^\circ + cos540^\circ .cos290^\circ }{cos1260^\circ }$

Exercise 7: Prove the equations
A) $\frac{1}{2}(cos \alpha + \sqrt{3}sin \alpha ) = sin(30^\circ + \alpha)$
B) cosα - √3sinα = 2cos(60° + α).
C) tg20° + tg23° + tg20°.tg25° = 1.
D) tg20° + tg40° + √3tg20°.tg40° = √3.
E) sin(α + β).cos(α - β) = sin²α - sin²β.
F) cos(α + β).cos(α - β) = cos²α - cos²β.
G) $sin^2\left(\alpha - \frac{\pi}{6} \right) + sin^2\left(\alpha + \frac{\pi}{6} \right) - sin^2 \alpha = \frac{1}{2}$