Educational

Prove that cos A + cos 3A + cos 5A + cos 7A = 4 cos A cos 2A cos 4A Prove that: \[ \cos A + \cos 3A + \cos 5A + \cos 7A = 4\cos A \cos 2A \cos 4A \] Solution Group the terms: \[ (\cos A + \cos 7A) + (\cos […]

Prove that cos 3A + cos 5A + cos 7A + cos 15A = 4 cos 4A cos 5A cos 6A Prove that: \[ \cos 3A + \cos 5A + \cos 7A + \cos 15A = 4\cos 4A \cos 5A \cos 6A \] Solution Group the terms: \[ (\cos 3A + \cos 15A) + (\cos

Prove that sin 47° + cos 77° = cos 17° Prove that: \[ \sin 47^\circ + \cos 77^\circ = \cos 17^\circ \] Solution Convert cosine into sine form using: \[ \cos \theta = \sin(90^\circ-\theta) \] \[ \cos 77^\circ = \sin 13^\circ \] Therefore, \[ \sin 47^\circ + \cos 77^\circ = \sin 47^\circ + \sin 13^\circ

Prove that sin 65° + cos 65° = √2 cos 20° Prove that: \[ \sin 65^\circ + \cos 65^\circ = \sqrt{2}\cos 20^\circ \] Solution Using the identity: \[ \sin \theta + \cos \theta = \sqrt{2}\cos(45^\circ-\theta) \] Taking \[ \theta = 65^\circ \] Then, \[ \sin 65^\circ + \cos 65^\circ = \sqrt{2}\cos(45^\circ-65^\circ) \] \[ = \sqrt{2}\cos(-20^\circ)

Prove that cos(π/4 + x) + cos(π/4 − x) = √2 cos x Prove that: \[ \cos\left(\frac{\pi}{4}+x\right) + \cos\left(\frac{\pi}{4}-x\right) = \sqrt{2}\cos x \] Solution Using the identity: \[ \cos A + \cos B = 2\cos\frac{A+B}{2}\cos\frac{A-B}{2} \] Taking \[ A=\frac{\pi}{4}+x, \qquad B=\frac{\pi}{4}-x \] Then, \[ \cos\left(\frac{\pi}{4}+x\right) + \cos\left(\frac{\pi}{4}-x\right) \] \[ = 2\cos\frac{\left(\frac{\pi}{4}+x\right)+\left(\frac{\pi}{4}-x\right)}{2} \cos\frac{\left(\frac{\pi}{4}+x\right)-\left(\frac{\pi}{4}-x\right)}{2} \] \[ =

Prove that cos(3π/4 + x) − cos(3π/4 − x) = −√2 sin x Prove that: \[ \cos\left(\frac{3\pi}{4}+x\right) – \cos\left(\frac{3\pi}{4}-x\right) = -\sqrt{2}\sin x \] Solution Using the identity: \[ \cos A – \cos B = -2\sin\frac{A+B}{2}\sin\frac{A-B}{2} \] Taking \[ A=\frac{3\pi}{4}+x, \qquad B=\frac{3\pi}{4}-x \] Then, \[ \cos\left(\frac{3\pi}{4}+x\right) – \cos\left(\frac{3\pi}{4}-x\right) \] \[ = -2\sin\frac{\left(\frac{3\pi}{4}+x\right)+\left(\frac{3\pi}{4}-x\right)}{2} \sin\frac{\left(\frac{3\pi}{4}+x\right)-\left(\frac{3\pi}{4}-x\right)}{2} \] \[ =

Prove that sin 51° + cos 81° = cos 21° Prove that: \[ \sin 51^\circ + \cos 81^\circ = \cos 21^\circ \] Solution Convert cosine into sine form using: \[ \cos \theta = \sin(90^\circ-\theta) \] \[ \cos 81^\circ = \sin 9^\circ \] Therefore, \[ \sin 51^\circ + \cos 81^\circ = \sin 51^\circ + \sin 9^\circ

Prove that sin 80° − cos 70° = cos 50° Prove that: \[ \sin 80^\circ – \cos 70^\circ = \cos 50^\circ \] Solution Convert cosine into sine form using: \[ \cos \theta = \sin(90^\circ-\theta) \] \[ \cos 70^\circ = \sin 20^\circ \] Therefore, \[ \sin 80^\circ – \cos 70^\circ = \sin 80^\circ – \sin 20^\circ

Prove that cos(π/12) − sin(π/12) = 1/√2 Prove that: \[ \cos \frac{\pi}{12} – \sin \frac{\pi}{12} = \frac{1}{\sqrt{2}} \] Solution Using the identity: \[ \cos \theta – \sin \theta = \sqrt{2}\cos\left(\theta+ \frac{\pi}{4}\right) \] Taking \[ \theta = \frac{\pi}{12} \] Then, \[ \cos \frac{\pi}{12} – \sin \frac{\pi}{12} = \sqrt{2}\cos\left(\frac{\pi}{12}+\frac{\pi}{4}\right) \] \[ = \sqrt{2}\cos\left(\frac{\pi}{12}+\frac{3\pi}{12}\right) \] \[ = \sqrt{2}\cos\frac{4\pi}{12}

Prove that sin(5π/18) − cos(4π/9) = √3 sin(π/9) Prove that: \[ \sin \frac{5\pi}{18} – \cos \frac{4\pi}{9} = \sqrt{3}\sin \frac{\pi}{9} \] Solution Convert the cosine term into sine form using: \[ \cos \theta = \sin\left(\frac{\pi}{2}-\theta\right) \] \[ \cos \frac{4\pi}{9} = \sin\left(\frac{\pi}{2}-\frac{4\pi}{9}\right) = \sin\frac{\pi}{18} \] Therefore, \[ \sin \frac{5\pi}{18} – \cos \frac{4\pi}{9} = \sin \frac{5\pi}{18} – \sin