Ravi Kant Kumar

Prove that: sin(4π/9 + θ) cos(π/9 + θ) − cos(4π/9 + θ) sin(π/9 + θ) = √3/2 Question Prove that: \[ \sin\left(\frac{4\pi}{9}+\theta\right)\cos\left(\frac{\pi}{9}+\theta\right) – \cos\left(\frac{4\pi}{9}+\theta\right)\sin\left(\frac{\pi}{9}+\theta\right) = \frac{\sqrt{3}}{2} \] Proof Consider the left-hand side: \[ \sin\left(\frac{4\pi}{9}+\theta\right)\cos\left(\frac{\pi}{9}+\theta\right) – \cos\left(\frac{4\pi}{9}+\theta\right)\sin\left(\frac{\pi}{9}+\theta\right) \] Using the identity: \[ \sin A\cos B-\cos A\sin B=\sin(A-B) \] Let \[ A=\frac{4\pi}{9}+\theta, \qquad B=\frac{\pi}{9}+\theta \] Then, \[

Prove that: sin(π/3 − x) cos(π/6 + x) + cos(π/3 − x) sin(π/6 + x) = 1 Question Prove that: \[ \sin\left(\frac{\pi}{3}-x\right)\cos\left(\frac{\pi}{6}+x\right) + \cos\left(\frac{\pi}{3}-x\right)\sin\left(\frac{\pi}{6}+x\right) =1 \] Proof Consider the left-hand side: \[ \sin\left(\frac{\pi}{3}-x\right)\cos\left(\frac{\pi}{6}+x\right) + \cos\left(\frac{\pi}{3}-x\right)\sin\left(\frac{\pi}{6}+x\right) \] Using the identity: \[ \sin A\cos B+\cos A\sin B=\sin(A+B) \] Let \[ A=\frac{\pi}{3}-x, \qquad B=\frac{\pi}{6}+x \] Then, \[ =

Prove that: (cos 8° − sin 8°)/(cos 8° + sin 8°) = tan 37° Question Prove that: \[ \frac{\cos 8^\circ-\sin 8^\circ} {\cos 8^\circ+\sin 8^\circ} = \tan 37^\circ \] Proof Consider the left-hand side: \[ \frac{\cos 8^\circ-\sin 8^\circ} {\cos 8^\circ+\sin 8^\circ} \] Divide numerator and denominator by \[ \cos 8^\circ \] \[ = \frac{\frac{\cos 8^\circ}{\cos 8^\circ}-\frac{\sin

Prove that: (cos 9° + sin 9°)/(cos 9° − sin 9°) = tan 54° Question Prove that: \[ \frac{\cos 9^\circ+\sin 9^\circ} {\cos 9^\circ-\sin 9^\circ} = \tan 54^\circ \] Proof Consider the left-hand side: \[ \frac{\cos 9^\circ+\sin 9^\circ} {\cos 9^\circ-\sin 9^\circ} \] Divide numerator and denominator by \[ \cos 9^\circ \] \[ = \frac{\frac{\cos 9^\circ}{\cos 9^\circ}+\frac{\sin

Prove that: (cos 11° + sin 11°)/(cos 11° − sin 11°) = tan 56° Question Prove that: \[ \frac{\cos 11^\circ+\sin 11^\circ} {\cos 11^\circ-\sin 11^\circ} = \tan 56^\circ \] Proof Consider the left-hand side: \[ \frac{\cos 11^\circ+\sin 11^\circ} {\cos 11^\circ-\sin 11^\circ} \] Divide numerator and denominator by \[ \cos 11^\circ \] \[ = \frac{\frac{\cos 11^\circ}{\cos 11^\circ}+\frac{\sin

Prove that: (tan A + tan B)/(tan A − tan B) = sin(A+B)/sin(A−B) Question Prove that: \[ \frac{\tan A+\tan B}{\tan A-\tan B} = \frac{\sin(A+B)}{\sin(A-B)} \] Proof Consider the left-hand side: \[ \frac{\tan A+\tan B}{\tan A-\tan B} \] Using \[ \tan \theta=\frac{\sin\theta}{\cos\theta} \] we get: \[ = \frac{\frac{\sin A}{\cos A}+\frac{\sin B}{\cos B}} {\frac{\sin A}{\cos A}-\frac{\sin B}{\cos

Prove that: cos 7π/12 + cos π/12 = sin 5π/12 − sin π/12 Question Prove that: \[ \cos \frac{7\pi}{12}+\cos \frac{\pi}{12} = \sin \frac{5\pi}{12}-\sin \frac{\pi}{12} \] Proof Consider the left-hand side: \[ \cos \frac{7\pi}{12}+\cos \frac{\pi}{12} \] Using the identity: \[ \cos C+\cos D = 2\cos\frac{C+D}{2}\cos\frac{C-D}{2} \] Let \[ C=\frac{7\pi}{12}, \qquad D=\frac{\pi}{12} \] Then, \[ \cos \frac{7\pi}{12}+\cos

If cos A = −12/13 and cot B = 24/7, Find sin(A+B), cos(A+B) and tan(A+B) Question If \[ \cos A=-\frac{12}{13} \] and \[ \cot B=\frac{24}{7} \] where A lies in the second quadrant and B lies in the third quadrant, find: (i) \(\sin(A+B)\) (ii) \(\cos(A+B)\) (iii) \(\tan(A+B)\) Solution Given: \[ \cos A=-\frac{12}{13} \] Using \[

Evaluate cos 80° cos 20° + sin 80° sin 20° Question Evaluate: \[ \cos 80^\circ \cos 20^\circ+\sin 80^\circ \sin 20^\circ \] Solution Using the identity: \[ \cos A \cos B+\sin A \sin B=\cos(A-B) \] Here, \[ A=80^\circ,\qquad B=20^\circ \] Therefore, \[ \cos 80^\circ \cos 20^\circ+\sin 80^\circ \sin 20^\circ \] \[ =\cos(80^\circ-20^\circ) \] \[ =\cos 60^\circ

Evaluate sin 36° cos 9° + cos 36° sin 9° Question Evaluate: \[ \sin 36^\circ \cos 9^\circ+\cos 36^\circ \sin 9^\circ \] Solution Using the identity: \[ \sin A \cos B+\cos A \sin B=\sin(A+B) \] Here, \[ A=36^\circ,\qquad B=9^\circ \] Therefore, \[ \sin 36^\circ \cos 9^\circ+\cos 36^\circ \sin 9^\circ \] \[ =\sin(36^\circ+9^\circ) \] \[ =\sin 45^\circ