Exercise 3.3

sin² π / 6 + cos² 3 π - tan² 1-4 2 π = 2. 2sin 2 π + cosec² 27 3cos6 3 2 π π = 3. 2 25cot cosec 3tan 66 6 6 π π π+ + = 4. 2 2 232sin 2cos 2sec 104 4 3 π π π+ + =

Find the value of:

(i) $\sin 75^\circ$

(ii) $\tan 15^\circ$

cos cos sin sin sin ( )4 4 4 4x y x y x yπ π π π       − − − − − = +               7. πtan 1 tan4 π 1 tantan 4 x x xx   +   +  =  −   −   8. 2cos ( ) cos ( ) cot sin ( ) cos 2

x x π + − =π π − +   

Prove that $\cos \left(\dfrac{3\pi}{2} + x\right) \cos(2\pi + x) \left[ \cot \left(\dfrac{3\pi}{2} - x\right) + \cot(2\pi + x) \right] = 1$

Show that $sin((n+1)x)sin((n+2)x)+cos((n+1)x)cos((n+2)x)=cos x$.

sin² 6x - sin² 4x = sin 2x sin 10x 13. cos² 2x - cos² 6x = sin 4 x sin 8x

sin² x + 2 sin 4x + sin 6x = 4 cos² x sin 4x

Prove that $\cot 4x (\sin 5x + \sin 3x) = \cot x (\sin 5x - \sin 3x)$