2) sin⁶x + cos⁶x = 1 - 3 tan²x cos²x
3) (cosecx - sinx)(secx - cosx)(tanx + cotx)= 1
4) cosecx (secx - 1) - cotx (1- cosx)= tanx - sinx.
5) (1- sinx cosx)/{cosx(secx - cosecx) . (Sin²x - cos²x)/(sin³x + cos³x) = sinx
6) tanx/(1 - cotx ) + cotx/(1 - tanx) = (secx cosecx + 1)
7) (sin³x + cos³x)/(sinx + cosx) + (sin³x - cos³x)/(sinx - cosx)= 2
8) (secx sec y + tanx tany)² - (secx tany + tanx secy)² = 1
9) cosx/(1- sinx) = (1+ cosx + sinx)/(1- sinx + cosx).
10) √{(1- sinx)/(1+ sinx)} + √{(1+ sinx)/(1- sinx)} = -2/cosx
11) tan³x/(1+ tan²x) + cot³x/(1+ cot²x) = (1- 2 sin²x cos²x)/(sinx cosx)
12) 1 - sin²x/(1+ cotx) - cos²x/(1+ tanx)= sinx cosx.
13) {1/(sec²x - cos²x) + 1/(cosec²x - sin²x)} sin²x cos²x = (1 - sin²x cos²x)/(2+ sin²x cos²x).
14) (1+ tanx tan y)²+ (tan x - tan y)²= sec²x sec²y.
15) (1+ cotx + tanx)(sinx - cosx)/(sec³x - cosec³x)= sin²x cos²x.
16) (2 sinx cosx- cosx)/(1- sinx + sin²x - cos²x)= cotx.
17) cosx(tanx +2)(2 tanx + 1)= 2 secx + 5 sinx.
18) sin⁸x - cos⁸x = (sin²x - cos²x)(1- 2 sin²x cos²x)
19) cot⁴x + cot²x= cosec⁴x - cosec²x.
20) 2 sec²x - sec⁴x - 2 cosec²x + cosec⁴x = cot⁴x - tan⁴x.
21) (sinx + cosecx)² +(cosx+ secx)² = tan²x +cot²x+ 7
22) (1+ cotx- cosecx)(1+ tanx + secx)= 2
23) (tanx + secx- 1)/(tanx - secx +1)= (1+ sin)/cosx
24) 3(sinx - cosx)⁴+ 6(sinx + cosx)² + 4(sin⁶x+ cos⁶x) - 13= 0
Height and Distances
1) A man standing on the bank of a river observes that the angle of elevation of a tree on the opposite bank is 60°. When he moves 50m away from the bank, he finds the angle of elevation to be 30°. Find:
A) the width of the river. 25m
B) the height of the tree. 43.3m
2) A telegraph pole is 8m high. Its shadow is 8√3m in length. Find the elevation of the sun. 30°
3) The angle of elevation of the top of a tower at a distance of x m. from its foot on a horizontal plane is found to be 30°. If the height of the tower be 70m, find x. 121.24m
4) A tree is broken by the wind. The top struck the ground at an angle of 30° and at a distance of 30 m from the foot. Find the whole height of the tree. 57.96
5) The angles of elevation of the top of a tower from two points on the ground at distance a metres and b metres from the base of the tower and in the same straight line with it are complementary. Prove that the height of the tower is √(ab) metres.
6) Two pillars of equal height stand on either side of a roadway which is 120m wide. At a point in the road between pillars, the elevations of the pillars are 60° and 30°. Find the height of each pillar and the position of the point. 51.96m, 30m from the pillar with elevation 60°
7) A man on the top of a vertical observation tower observes a car moving at a uniform speed coming directly towards it. If it takes 12 minutes for the angle of depression to change from 30° to 45°, how soon after this will the car reach the observation tower? 16.39m
8) A man in a boat rowing away from a tower 150m high takes 2 minutes to change the angle of elevation of the top of the tower from 60° to 45°. Find the speed of the boat. 0.53 m/s
9) A car is travelling at 50 m/s along a road in inclined at 30° to horizontal. In what time will the vertical height of the car increase by 50 m ? 2 sec
10) A boy standing on the ground and flying a kite with 100m of string at an elevation of 30°. Another boy is standing on the roof of a 20m high building and is flying his kite at an elevation of 45°. Both the boys are on opposite sides of both the kites. Find the length of the string that the second boy must have so that the two kites meet. 42.32
11) An aeroplane when flying at a height of 4000 m from the ground passes vertically above another aeroplane at an instant when the angle of elevation of the two planes from the same point of the ground are 60° and 45° respectively. Find the vertical distance between the aeroplanes at that instant. 1687.86
12) The angles of elevation of an aeroplane at two consecutive kilometre posts respectively are a and b. Find the height of the aeroplane above the ground taking it to be between the two kilometre stone. (tan a tan b)/(tan a + tan b)
13) A tower subtends an angle a at a point on the same level as the foot of the tower and at a second point h metres above the first, the depression of the foot of the tower is b. Show that the height of the tower is h tan a tan b.
14) From a window (h metre high above the ground) of a house in a street, the angle of elevation and depression of the top and the foot of another house on opposite side of the street are a and b respectively. show that the height of the opposites house is h(1+ tan a cot b).
15) The angles of elevation of the Summit of a hill from the top and bottom of a tower are 45° and 60° respectively. If the height of the tower is h m. Prove that the height of the hill is {√3(1+√3)}/2 metres.
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