(i) Let us consider the LHS:
tan 225° cot 405° + tan 765° cot 675°
tan (90° × 2 + 45°) cot (90° × 4 + 45°) + tan (90° × 8 + 45°) cot (90° × 7 + 45°)
As we know that when n is odd, cot → tan.
tan 45° cot 45° + tan 45° [-tan 45°]
tan 45° cot 45° – tan 45° tan 45°
1 × 1 – 1 × 1
1 – 1
0 = RHS
∴ LHS = RHS
Thus proved.
(ii) sin 8π/3 cos 23π/6 + cos 13π/3 sin 35π/6 = 1/2
Let us consider the LHS:
sin 8π/3 cos 23π/6 + cos 13π/3 sin 35π/6
sin 480° cos 690° + cos 780° sin 1050°
sin (90° × 5 + 30°) cos (90° × 7 + 60°) + cos (90° × 8 + 60°) sin (90° × 11 + 60°)
As we know that when n is odd, sin → cos and cos → sin.
cos 30° sin 60° + cos 60° [-cos 60°]
√3/2 × √3/2 – 1/2 × 1/2
3/4 – 1/4
2/4
1/2
= RHS
∴ LHS = RHS
Thus proved.
(iii) Let us consider the LHS:
cos 24° + cos 55° + cos 125° + cos 204° + cos 300°
cos 24° + cos (90° × 1 – 35°) + cos (90° × 1 + 35°) + cos (90° × 2 + 24°) + cos (90° × 3 + 30°)
As we know that when n is odd, cos → sin.
cos 24° + sin 35° – sin 35° – cos 24° + sin 30°
0 + 0 + 1/2
1/2
= RHS
∴ LHS = RHS
Thus proved.
(iv) Let us consider the LHS:
tan (-125°) cot (-405°) – tan (-765°) cot (675°)
As we know that tan (-x) = -tan (x) and cot (-x) = -cot (x).
[-tan (225°)] [-cot (405°)] – [-tan (765°)] cot (675°)
tan (225°) cot (405°) + tan (765°) cot (675°)
tan (90° × 2 + 45°) cot (90° × 4 + 45°) + tan (90° × 8 + 45°) cot (90° × 7 + 45°)
tan 45° cot 45° + tan 45° [-tan 45°]
1 × 1 + 1 × (-1)
1 – 1
0
= RHS
∴ LHS = RHS
Thus proved.
(v) Let us consider the LHS:
cos 570° sin 510° + sin (-330°) cos (-390°)
As we know that sin (-x) = -sin (x) and cos (-x) = +cos (x).
cos 570° sin 510° + [-sin (330°)] cos (390°)
cos 570° sin 510° – sin (330°) cos (390°)
cos (90° × 6 + 30°) sin (90° × 5 + 60°) – sin (90° × 3 + 60°) cos (90° × 4 + 30°)
As we know that cos is negative at 90° + θ i.e. in Q2 and when n is odd, sin → cos and cos → sin.
-cos 30° cos 60° – [-cos 60°] cos 30°
-cos 30° cos 60° + cos 60° cos 30°
0
= RHS
∴ LHS = RHS
Thus proved.
(vi) tan 11π/3 – 2 sin 4π/6 – 3/4 cosec2 π/4 + 4 cos2 17π/6 = (3 – 4√3)/2
Let us consider the LHS:
tan 11π/3 – 2 sin 4π/6 – 3/4 cosec2 π/4 + 4 cos2 17π/6
tan (11 × 180°)/3 – 2 sin (4 × 180°)/6 – 3/4 cosec2 180°/4 + 4 cos2 (17 × 180°)/6
tan 660° – 2 sin 120° – 3/4 (cosec 45°)2 + 4 (cos 510°)2
tan (90° × 7 + 30°) – 2 sin (90° × 1 + 30°) – 3/4 [cosec 45°]2 + 4 [cos (90° × 5 + 60°)]2
As we know that tan and cos is negative at 90° + θ i.e. in Q2 and when n is odd, tan → cot, sin → cos and cos → sin.
[-cot 30°] – 2 cos 30° – 3/4 [cosec 45°]2 + [-sin 60°]2
– cot 30° – 2 cos 30° – 3/4 [cosec 45°]2 + [sin 60°]2
-√3 – 2√3/2 – 3/4 (√2)2 + 4 (√3/2)2
-√3 – √3 – 6/4 + 12/4
(3 – 4√3)/2
= RHS
∴ LHS = RHS
Thus proved.
(vii) 3 sin π/6 sec π/3 – 4 sin 5π/6 cot π/4 = 1
Let us consider the LHS:
3 sin π/6 sec π/3 – 4 sin 5π/6 cot π/4
3 sin 180°/6 sec 180°/3 – 4 sin 5(180°)/6 cot 180°/4
3 sin 30° sec 60° – 4 sin 150° cot 45°
3 sin 30° sec 60° – 4 sin (90° × 1 + 60°) cot 45°
As we know that when n is odd, sin → cos.
3 sin 30° sec 60° – 4 cos 60° cot 45°
3 (1/2) (2) – 4 (1/2) (1)
3 – 2
1
= RHS
∴ LHS = RHS
Thus proved.