(i) tan(-225°) = -(tan 225°) 

= -(tan(180° + 45°)) 

= – tan 45° = – 1 

cot(-405°) = -(cot 405°) 

= – cot(360° + 45°) [∵ For 360° + 45° no change in T-ratio.] 

= -cot 45° 

= -1 

tan(-765°) = -tan 765° 

= -tan(2 x 360° + 45°) 

= -tan 45° 

= -1 

cot 675° = cot (360°+ 315°) 

= cot 315° 

= cot(360° – 45°) 

= -cot 45° 

= -1 

LHS = tan(-225°) cot(-405°) – tan(-765°) cot(675°) 

= (-1) (-1) – (-1) (-1) 

= 1 – 1

= 0 = RHS. 

Hence proved.

(ii) 2 sin² \(\frac{\pi}{6}\) + cosec² \(\frac{7\pi}{6}\) cos² \(\frac{\pi}{3}\)= \(\frac{3}{2}\)

LHS = 2 sin² \(\frac{\pi}{6}\) + cosec² \(\frac{7\pi}{6}\) cos² \(\frac{\pi}{3}\)= \(\frac{3}{2}\)

[∵ \(\frac{7\pi}{6}\)= 210°, 210° = 180° + 30°. For 180° + 30° no change in T-ratio. 210° lies in 3rd quadrant, cosec θ is negative.] 

= 2(sin\(\frac{\pi}{6}\))² + (cosec (180° + 30°))² (cos\(\frac{\pi}{3}\))²

= 2(\(\frac{1}{2}\))² + (-cosec 30°)².(\(\frac{1}{2}\))² 

= 2 x \(\frac{1}{4}+(-2)^2\frac{1}{4}\) 

= \(\frac{2}{4}+\frac{4}{4}=\frac{6}{4}\)

= \(\frac{6}{4}\)

= \(\frac{3}{2}\)

= RHS

(iii) sec(\(\frac{3\pi}{2}-\theta\)) = sec (270° – θ) = -cosec θ 

[∵ For 270° – θ change T-ratio. So add ‘co’ infront ‘sec’, it becomes ‘cosec’] 

sec(θ – \(\frac{5\pi}{2}\)) = sec(-(\(\frac{5\pi}{2}\) - θ))

= sec(\(\frac{5\pi}{2}\) – θ) [∵ sec(-θ) = θ] 

= sec(450° – θ) 

= sec (360° + (90° – θ))

= sec (90° – θ) 

= cosec θ 

[∵ For 90° – θ change in T-ratio. So add ‘co’ in front of ‘sec’ it becomes ‘cosec’] 

tan(\(\frac{5\pi}{2}\) + θ) = tan(450° + θ) 

[∵ For 90° + θ, change in T-ratio. So add ‘co’ in front of ‘tan’ it becomes ‘cot’] 

= tan (360° + (90° + θ)) 

= tan (90° + θ) 

= -cot θ

 tan(\(\theta-\frac{5\pi}{2}\)) = tan(-(\(\frac{5\pi}{2}\) - θ))

= -  tan (\(\frac{5\pi}{2}\) - θ) [∵ tan(-θ) = -tan θ] 

= -tan(450° – θ) 

= -tan(360° + (90° – θ)) 

= -tan(90° – θ) 

= -cot θ 

LHS = sec(\(\frac{3\pi}{2}-\theta\)) sec(\(\theta-\frac{5\pi}{2}\)) + tan(\(\frac{5\pi}{2}+\theta\)) tan(\(\theta-\frac{5\pi}{2}\))

= -cosec θ (cosec θ) + (-cot θ) (-cot θ) 

= -cosec² θ + cot² θ 

= -(1 + cot² θ) + cot² θ [∵ 1 + cot² θ = cosec² θ] 

= -1 

= RHS