Trigonometric identities: Problems with Solutions

By Prof. Hernando Guzman Jaimes (University of Zulia - Maracaibo, Venezuela)

Related topics:
Trigonometry
Trigonometric equations

Problem 1
Which of the following trigonometric identities is true?
Problem 2
Which of the following trigonometric identities is true?
Problem 3
Which of the following trigonometric identities is true?
Problem 4
Which of the following trigonometric identities is true?
Problem 5
Which of the following trigonometric identities is true?
Problem 6
Which of the following trigonometric identities is true?
Problem 7
Which of the following trigonometric identities is true?
Problem 8
Find the values of the sine, cosine and tangent of $15^{\circ }$.
Hint: $15^{\circ }=45^{\circ }-30^{\circ }$

A) $\sin 15^{\circ }=\frac{\sqrt{2}}{4}\left( \sqrt{3}+1\right) \quad \cos 15^{\circ }=\frac{\sqrt{2}}{4}\left( \sqrt{3}-1\right) \quad \tan 15^{\circ }=2-\sqrt{3}$

B) $\sin 15^{\circ }=\frac{\sqrt{2}}{4}\left( \sqrt{3}-1\right) \quad \cos 15^{\circ }=\frac{\sqrt{2}}{4}\left( \sqrt{3}+1\right) \quad \tan 15^{\circ }=2+\sqrt{3}$

C) $\sin 15^{\circ }=\frac{\sqrt{2}}{4}\left( \sqrt{3}-1\right) \quad \cos 15^{\circ }=\frac{\sqrt{2}}{4}\left( \sqrt{3}+1\right) \quad \tan 15^{\circ }=2-\sqrt{3}$

D) $\sin 15^{\circ }=\frac{\sqrt{2}}{4}\left( \sqrt{3}+1\right) \quad \cos 15^{\circ }=\frac{\sqrt{2}}{4}\left( \sqrt{3}-1\right) \quad \tan 15^{\circ }=2+\sqrt{3}$
Problem 9
Find the values of the sine, cosine and tangent of $75^{\circ }$
Hint: $75^{\circ }=90^{\circ }-15^{\circ }$

A) $\sin 75^{\circ }=\frac{\sqrt{2}}{4}\left( \sqrt{3}-1\right) \quad \cos 75^{\circ }=\frac{\sqrt{2}}{4}\left( \sqrt{3}+1\right) \quad \tan 75^{\circ }=\frac{\left( \sqrt{3}+1\right) ^{2}}{2}$

B) $\sin 75^{\circ }=\frac{\sqrt{2}}{4}\left( \sqrt{3}+1\right) \quad \cos 75^{\circ }=\frac{\sqrt{2}}{4}\left( \sqrt{3}-1\right) \quad \tan 75^{\circ }=\frac{\left( \sqrt{3}-1\right) ^{2}}{2}$

C) $\sin 75^{\circ }=\frac{\sqrt{2}}{4}\left( \sqrt{3}-1\right) \quad \cos 75^{\circ }=\frac{\sqrt{2}}{4}\left( \sqrt{3}+1\right) \quad \tan 75^{\circ }=\frac{\left( \sqrt{3}-1\right) ^{2}}{2}$

D) $\sin 75^{\circ }=\frac{\sqrt{2}}{4}\left( \sqrt{3}+1\right) \quad \cos 75^{\circ }=\frac{\sqrt{2}}{4}\left( \sqrt{3}-1\right) \quad \tan 75^{\circ }=\frac{\left( \sqrt{3}+1\right) ^{2}}{2}$
Problem 10
$\sin \left( \alpha +\beta \right) +\sin \left( \alpha -\beta \right) =$

Problem 11
$\sin \left( \alpha +\beta \right) -\sin \left( \alpha -\beta \right) =$
Problem 12
$\cos \left( \alpha +\beta \right) +\cos\left( \alpha -\beta \right) =$
Problem 13
$\cos \left( \alpha +\beta \right) -\cos \left( \alpha -\beta \right) =$
Problem 14
$\frac{\tan \left( \alpha +\beta \right) -\tan \alpha }{1+\tan \left( \alpha +\beta \right) \tan \alpha }=$
Problem 15
Evaluate the trigonometric expression:
$\left( \sin \alpha \cos \beta -\cos \alpha \sin \beta \right)^{2}+\left( \cos \alpha \cos \beta +\sin \alpha \sin \beta \right)^{2}=$
Problem 16
$\cot \left( \alpha +\beta \right) =$
Problem 17
Which of the following trigonometric identities is true?
Problem 18
$\sin \frac{1}{2}\theta =$
Problem 19
$\cos \frac{1}{2}\theta =$
Problem 20
$\tan \frac{1}{2}\theta =$
Problem 21
Which of the following trigonometric identities is true?
Problem 22
Simplify the following trigonometric expression:
$\sin \left( \theta+30^{\circ }\right) +\cos \left( \theta +60^{\circ}\right)=$
Problem 23
Simplify: $\frac{1-\tan ^{2}\frac{1}{2}x}{1+\tan ^{2}\frac{1}{2}x}=$
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