Ravi Kant Kumar

Prove that cos 20° cos 100° + cos 100° cos 140° − cos 140° cos 20° = −3/4 Prove that: \[ \cos20^\circ\cos100^\circ + \cos100^\circ\cos140^\circ – \cos140^\circ\cos20^\circ = -\frac{3}{4} \] Solution Use the identity: \[ \cos A \cos B = \frac{1}{2} \left[ \cos(A+B)+\cos(A-B) \right] \] First term: \[ \cos20^\circ\cos100^\circ = \frac{1}{2} \left[ \cos120^\circ+\cos(-80^\circ) \right] \] \[

Prove that sin 3A + sin 2A − sin A = 4 sin A cos(A/2) cos(3A/2) Prove that: \[ \sin 3A + \sin 2A – \sin A = 4\sin A \cos\frac{A}{2}\cos\frac{3A}{2} \] Solution Group the first and third terms: \[ (\sin 3A – \sin A) + \sin 2A \] Using the identity: \[ \sin A

Prove that sin A + sin 2A + sin 4A + sin 5A = 4 cos(A/2) cos(3A/2) sin 3A Prove that: \[ \sin A + \sin 2A + \sin 4A + \sin 5A = 4\cos\frac{A}{2}\cos\frac{3A}{2}\sin 3A \] Solution Group the terms: \[ (\sin A + \sin 5A) + (\sin 2A + \sin 4A) \] Using

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