4.5.21
1) sin(2tan⁻¹1/5 - tan⁻¹5/12). 0
2) tan[ sin⁻¹1/3 +cos⁻¹1/√3]. 5/√2
3) sin[2sin⁻¹1/√26+sin⁻¹12/13]. 1
4) tan[2tan⁻¹1/5 - π/4]. -7/17
5) tan⁻¹sin cos⁻¹√(2/3). π/6
6) tan(cos⁻¹4/5 + tan⁻¹2/3). 17/6
7) tan{1/2(tan⁻¹x + tan⁻¹1/x)}. N.s
8) sin{tan⁻¹(tan7π/6)+ cos⁻¹(cos 7π/3)} is
A) 0 B) -1 C) 1 D) none
9) cos⁻¹y + cos⁻¹(x/2 + 1/2 √(3-3x²), 1/2≤x≤1. π/3
9.5.21
B) Prove:
1) 2tan⁻¹2 + tan⁻¹3 = π+ tan⁻¹1/3
2) 2cos⁻¹2/√5+ cos⁻¹3/5 =sin⁻¹24/25
3) 4tan⁻¹1/5 - tan⁻¹1/239=π/4
4) cos⁻¹8/17 + cos⁻¹3/5 + cos⁻¹36/85= π
5) cos⁻¹a - sin⁻¹b = cos⁻¹[b√(1-a²) + a √(1-b²)].
6) tan⁻¹p/q - tan⁻¹{(p-q)/(p+q)} = π/4
7) tan⁻¹x + tan⁻¹y + tan⁻¹{(1-x-y-xy)/(1+x+y- xy)}=π/4
8) tan⁻¹{(2a- b)/b√3} + tan⁻¹{2b-a)/a√3}=π/3
9) cos{1/2 cos⁻¹(-1/9)}= 2/3
10) a cosx + b sinx =√(a²+b²) cos(x - tan⁻¹b/a)= √(a²+ b²) sin(x + tan⁻¹a/b)
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11) 2 tan⁻¹a + 2tan⁻¹b = sin⁻¹[{2(a+b)(1-ab)}/{(1+a²)(1+b²)}
12) tan(1/2 cos⁻¹a)= √{(1-a)/(1+a)}
13) cos⁻¹√(3/5)= 1/2 cos⁻¹1/5
14) 2cos⁻¹3/√13 + cot⁻¹16/63 + 1/2 cos⁻¹7/25=π
15) cos⁻¹√(2/3) - cos⁻¹(√6+1)/2√3}= π/6.
16) 2tan⁻¹[√50 - √18 - 1/√{3- 2√2)} = π/4
17) 2tan⁻¹{tan(π/4 - a) tan b/2} = cos⁻¹{(sin 2a + cos b)/(1+ sin 2a cos b)}
18) 2tan⁻¹{tana/2 tan (π/4 - b/2)} = tan⁻¹{(sina cos b)/(cos a+ sin b)
19) tan⁻¹1+ tan⁻¹[tan²(a+b) tan²(a-b)]= tan⁻¹[1/2 (cos 2a sec 2b + cos 2b sec 2a]
20) a³/2 cosec²(1/2 tan⁻¹a/b) + b³/2 sec²(1/2 tan⁻¹b/a)= (a+b)(a²+b²).
21) tan⁻¹{(3 sin 2x)/(5+ 3cos 2x)} + tan⁻¹(1/4 tanx) = x
22) tan⁻¹{(2 sin 2x)/(1+ 2cos 2x)} - 1/2 sin⁻¹{(3 sin 2x)/(5+ 4 cos 2x)}= x
C) SOLVE:
1) tan⁻x + tan⁻¹2x + tan⁻3x= π. 1
2) cot⁻¹x + sin⁻¹1/√5=π/4. 3
3) tan⁻¹{(x-1)/(x-2)} + tan⁻¹{(x+1)/(x+2)}=π/4. ±1/√2
4) cos⁻¹8/x + cos⁻¹5/x =π/2. 16
5) sin⁻x + sin⁻¹(1-x)= sin⁻¹√(1-x²). 0, 1/2
6) tan⁻¹{(x+1)/(x-1)}+ tan⁻¹{x-1)/x}= tan⁻¹(-7), (x≠0,1). No sol.
7) sec⁻¹x/a + sec⁻¹x/b = sec⁻¹a + sec⁻¹b. ab
8) sin⁻¹x - cos⁻¹x = sin⁻¹(3x-2). 1/2,1
9) cos⁻¹{(x²-1)/(x²+1)}+ tan⁻¹{2x/(x²+1)= 2π/3. √3, 2-√3
10) sin [2cos⁻¹cot(2tan⁻¹x)]=0. ±1, 1±√2, -1±√2
11) tan⁻¹{(x-1)/(x+1)} + tan⁻¹{2x-1)/(2x+1)}= tan⁻¹7/6. 2
* If sin⁻¹x + sin⁻¹y= π/2, prove 2(x² - xy +y²)= 1+ x⁴+y⁴
* If a= tan⁻¹{x√3/(2k -x)} and b=tan⁻¹{(2x-k)/k√3}. Show that one of the values of a- b is π/6.
* If tan⁻¹yz/xr + tan⁻¹zx/yr + tan⁻¹xy/zr = π/2, prove x² + y² + z² = r²
* If sin⁻¹x + sin⁻¹y= 2π/3, find the value of cos⁻¹x + cos⁻¹y. 1
* If cos⁻¹x/2 + cos⁻¹y/3= a, prove that 9x² - 12xy cos a + 4y²= 36 sin²a
* Show that the least value of (sin⁻¹x)³ + (cos⁻¹x)³ is π³/32.
* Prove tan⁻¹√(xr/yz) + tan⁻¹√(yr/zx) + tan⁻¹√(zr/xy)=π, when x+y+z= r
* If tan⁻¹x + tan⁻¹y + tan⁻¹z=π/2 and x+y+z= √3, show that, x=y=z
* If cos⁻¹x + cos⁻¹y + cos⁻¹z=π and x+y+z= 3/2, prove that, x=y=z
* If sin⁻¹x/a + sin⁻¹y/b = sin⁻¹c²/ab, then show b²x² + 2xy√(a²b² - c⁴) + a²y² = c⁴.
* If ax + b sec(tan⁻¹x)= c and ay + b sec(tan⁻¹y)= c, show that, (x+y)/(1-xy) = 2ac/(a² - c²).
* If A= sin⁻¹(sinx + siny) - sin⁻¹(sinx - siny) and sin²y + sin²x = 1/2 (0<x<π/2), then show that, cosA= cos 2y - cos 2x.
* If tan⁻¹√{(a²-x²)/(a²+x²)} + tan⁻¹{(b² - y²)/(b²+ y²)} = k/2, prove x⁴/a⁴ - (2x²y²cos k)/a²b² + y⁴/b⁴ = sin²k
* If {m tan(a-x)}/cos²x = n tan x/cos²(a-x), then show x= 1/2[a- tan⁻¹{(n-m)/(n+m)} tana]
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