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Mathematics
Prove that:
\[\sin\frac{x}{2}\sin\frac{7x}{2} + \sin\frac{3x}{2}\sin\frac{11x}{2} = \sin 2x \sin 5x .\]
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
Concept: undefined >> undefined
Prove that \[\cos x \cos \frac{x}{2} - \cos 3x \cos\frac{9x}{2} = \sin 7x \sin 8x\]
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
Concept: undefined >> undefined
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Prove that:
\[\frac{\sin A + \sin 3A}{\cos A - \cos 3A} = \cot A\]
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
Concept: undefined >> undefined
Prove that:
\[\frac{\sin 9A - \sin 7A}{\cos 7A - \cos 9A} = \cot 8A\]
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
Concept: undefined >> undefined
Prove that:
\[\frac{\sin A - \sin B}{\cos A + \cos B} = \tan\frac{A - B}{2}\]
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
Concept: undefined >> undefined
Prove that:
\[\frac{\sin A + \sin B}{\sin A - \sin B} = \tan \left( \frac{A + B}{2} \right) \cot \left( \frac{A - B}{2} \right)\]
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
Concept: undefined >> undefined
Prove that:
\[\frac{\cos A + \cos B}{\cos B - \cos A} = \cot \left( \frac{A + B}{2} \right) \cot \left( \frac{A - B}{2} \right)\]
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
Concept: undefined >> undefined
Prove that:
\[\frac{\cos 3A + 2 \cos 5A + \cos 7A}{\cos A + 2 \cos 3A + \cos 5A} = \frac{\cos 5A}{\cos 3A}\]
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
Concept: undefined >> undefined
Prove that:
\[\frac{\sin A + \sin 3A + \sin 5A}{\cos A + \cos 3A + \cos 5A} = \tan 3A\]
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
Concept: undefined >> undefined
Prove that:
\[\frac{\cos 4A + \cos 3A + \cos 2A}{\sin 4A + \sin 3A + \sin 2A} = \cot 3A\]
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
Concept: undefined >> undefined
Prove that:
\[\frac{\sin 3A + \sin 5A + \sin 7A + \sin 9A}{\cos 3A + \cos 5A + \cos 7A + \cos 9A} = \tan 6A\]
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
Concept: undefined >> undefined
Prove that:
\[\frac{\sin 5A - \sin 7A + \sin 8A - \sin 4A}{\cos 4A + \cos 7A - \cos 5A - \cos 8A} = \cot 6A\]
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
Concept: undefined >> undefined
Prove that:
\[\frac{\sin 5A \cos 2A - \sin 6A \cos A}{\sin A \sin 2A - \cos 2A \cos 3A} = \tan A\]
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
Concept: undefined >> undefined
Prove that:
\[\frac{\sin 11A \sin A + \sin 7A \sin 3A}{\cos 11A \sin A + \cos 7A \sin 3A} = \tan 8A\]
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
Concept: undefined >> undefined
Prove that:
\[\frac{\sin 3A \cos 4A - \sin A \cos 2A}{\sin 4A \sin A + \cos 6A \cos A} = \tan 2A\]
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
Concept: undefined >> undefined
Prove that:
\[\frac{\sin A \sin 2A + \sin 3A \sin 6A}{\sin A \cos 2A + \sin 3A \cos 6A} = \tan 5A\]
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
Concept: undefined >> undefined
Prove that:
\[\frac{\sin A + 2 \sin 3A + \sin 5A}{\sin 3A + 2 \sin 5A + \sin 7A} = \frac{\sin 3A}{\sin 5A}\]
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
Concept: undefined >> undefined
Prove that:
\[\frac{\sin \left( \theta + \phi \right) - 2 \sin \theta + \sin \left( \theta - \phi \right)}{\cos \left( \theta + \phi \right) - 2 \cos \theta + \cos \left( \theta - \phi \right)} = \tan \theta\]
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
Concept: undefined >> undefined
Prove that:
\[\sin \alpha + \sin \beta + \sin \gamma - \sin (\alpha + \beta + \gamma) = 4 \sin \left( \frac{\alpha + \beta}{2} \right) \sin \left( \frac{\beta + \gamma}{2} \right) \sin \left( \frac{\gamma + \alpha}{2} \right)\]
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
Concept: undefined >> undefined
Prove that:
cos (A + B + C) + cos (A − B + C) + cos (A + B − C) + cos (− A + B + C) = 4 cos A cos Bcos C
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
Concept: undefined >> undefined
