In triangle ABC, prove the following:
\[\frac{a - b}{a + b} = \frac{\tan \left( \frac{A - B}{2} \right)}{\tan \left( \frac{A + B}{2} \right)}\]
[3] Trigonometric Functions
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
In triangle ABC, prove the following:
\[\left( a - b \right) \cos \frac{C}{2} = c \sin \left( \frac{A - B}{2} \right)\]
[3] Trigonometric Functions
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
In triangle ABC, prove the following:
\[\frac{c}{a - b} = \frac{\tan\left( \frac{A}{2} \right) + \tan \left( \frac{B}{2} \right)}{\tan \left( \frac{A}{2} \right) - \tan \left( \frac{B}{2} \right)}\]
[3] Trigonometric Functions
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
In triangle ABC, prove the following:
\[\frac{c}{a + b} = \frac{1 - \tan \left( \frac{A}{2} \right) \tan \left( \frac{B}{2} \right)}{1 + \tan \left( \frac{A}{2} \right) \tan \left( \frac{B}{2} \right)}\]
[3] Trigonometric Functions
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
In triangle ABC, prove the following:
\[\frac{a + b}{c} = \frac{\cos \left( \frac{A - B}{2} \right)}{\sin \frac{C}{2}}\]
[3] Trigonometric Functions
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
In any triangle ABC, prove the following:
\[\sin \left( \frac{B - C}{2} \right) = \frac{b - c}{a} \cos\frac{A}{2}\]
[3] Trigonometric Functions
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
In triangle ABC, prove the following:
\[\frac{a^2 - c^2}{b^2} = \frac{\sin \left( A - C \right)}{\sin \left( A + C \right)}\]
[3] Trigonometric Functions
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
In triangle ABC, prove the following:
\[b \sin B - c \sin C = a \sin \left( B - C \right)\]
[3] Trigonometric Functions
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
In triangle ABC, prove the following:
\[a^2 \sin \left( B - C \right) = \left( b^2 - c^2 \right) \sin A\]
[3] Trigonometric Functions
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
In triangle ABC, prove the following:
\[\frac{\sqrt{\sin A} - \sqrt{\sin B}}{\sqrt{\sin A} + \sqrt{\sin B}} = \frac{a + b - 2\sqrt{ab}}{a - b}\]
[3] Trigonometric Functions
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
In triangle ABC, prove the following:
\[a \left( \sin B - \sin C \right) + \left( \sin C - \sin A \right) + c \left( \sin A - \sin B \right) = 0\]
[3] Trigonometric Functions
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
In triangle ABC, prove the following:
\[\frac{a^2 \sin \left( B - C \right)}{\sin A} + \frac{b^2 \sin \left( C - A \right)}{\sin B} + \frac{c^2 \sin \left( A - B \right)}{\sin C} = 0\]
[3] Trigonometric Functions
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
In triangle ABC, prove the following:
\[a^2 \left( \cos^2 B - \cos^2 C \right) + b^2 \left( \cos^2 C - \cos^2 A \right) + c^2 \left( \cos^2 A - \cos^2 B \right) = 0\]
[3] Trigonometric Functions
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
In triangle ABC, prove the following:
\[b \cos B + c \cos C = a \cos \left( B - C \right)\]
[3] Trigonometric Functions
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
In triangle ABC, prove the following:
\[\frac{\cos 2A}{a^2} - \frac{\cos 2B}{b^2} - \frac{1}{a^2} - \frac{1}{b^2}\]
[3] Trigonometric Functions
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
In triangle ABC, prove the following:
\[\frac{\cos^2 B - \cos^2 C}{b + c} + \frac{\cos^2 C - \cos^2 A}{c + a} + \frac{co s^2 A - \cos^2 B}{a + b} = 0\]
[3] Trigonometric Functions
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
In ∆ABC, prove that: \[a \sin\frac{A}{2} \sin \left( \frac{B - C}{2} \right) + b \sin \frac{B}{2} \sin \left( \frac{C - A}{2} \right) + c \sin \frac{C}{2} \sin \left( \frac{A - B}{2} \right) = 0\]
[3] Trigonometric Functions
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
In ∆ABC, prove that: \[\frac{b \sec B + c \sec C}{\tan B + \tan C} = \frac{c \sec C + a \sec A}{\tan C + \tan A} = \frac{a \sec A + b \sec B}{\tan A + \tan B}\]
[3] Trigonometric Functions
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
In triangle ABC, prove the following:
\[a \cos A + b\cos B + c \cos C = 2b \sin A \sin C = 2 c \sin A \sin B\]
[3] Trigonometric Functions
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
\[a \left( \cos B \cos C + \cos A \right) = b \left( \cos C \cos A + \cos B \right) = c \left( \cos A \cos B + \cos C \right)\]
[3] Trigonometric Functions
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
