# New PDF release: Arithmetic of Complex Manifolds By Barth W.P. (ed.), Lange H. (ed.)

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2 6. If sin""1^ ·+- cos~ 1 2x = -~ n where both sin - 1 a;and cos -1 (2a;) are acute, prove that 4x] (1 - x2) - x^(\ - 4x2) + 1 = 0 . 7. If s i n - 1 » = —ix for 0, write down the value of cos -1 a;. _ f a + x\ ,ί x\ π 8. Prove t h a t t a n " 1 -tan1 — = τ\a — x ) \a J 4 9. If Sin -1 a; = nn + (— l ) n a and Cos _1 a; = 2nn ± β, prove t h a t l tan (a + 2/9) = -x/f( ~ * 2 )· 10. Differentiate each of the following with respect to x: (i) sin" 1 (2a:), (ii) c o s " 1 ^ ) , (iii) t a n " 1 {(1 + x)/(l - x)}, (iv) t a n - 1 {(a + x)/(a — x)}, 1 (viii) sin" |cos(a; + — n \ i , (v) sin _ 1 { — I, (vi) tan _ 1 (sina;), (vii) t a n - 1 (2tanx), (ix) cos-^sin 2a;), (x) tan" 1 (e 2 a : + 1).

By eliminating x and y show that there is only one real value of z satisfying the equations zx - 3y - 19 = 0, 3x4- ly 4- 3z = 0, 2x 4- zy 4- 11 = 0. Solve the equations for this value of z. ) 37. Show that x and x + y + z are factors of the determinant (y + zf y* z* (2 + x)2 2* (x + yf I Hence, or otherwise, evaluate the determinant as a product of linear factors. ) LINEAR EQUATIONS AND D E T E R M I N A N T S 39 38. Show that the simultaneous equations x + ky = 2, y - kz = 3, kx — z ~ — 5, where & is real, have a unique solution provided k Φ — 1, and that if k= — 1 the equations are consistent.

Sinh x + sinht/ = 2 sinh — ^ — cosh — = — . 7. cosh a; cosh y = J {cosh (a; + y) + cosh (a: — y)}. 8. tanh 2 a: + sech2a; = 1. n J. i_ i//cosh 2a:- 1\ r 10. cosh 2 (a: + 2/) — cosh 2 (a: — y) Ξ= sinh 2 a: sinh 2y. 11. If sinh a; = 5/12, calculate cosh a: and tanha:. 3 12. If tanha: = -r-, calculate secha: and sinh a:. 13. 5. 14. Solve the equation 4 tanha: = cotha\ 15. Solve the equation 7 cosecha: + 2 cotha: = 6. 16. Draw sketch graphs of (i) tanha:, (ii) cosecha:, (iii) secha:. 17. Differentiate each of the following: (i) cosh (2a: -f 5), (ii) a:sinha:, (iii) sinh 2a: cosh 3a*, (iv) ex tanha:, (v) cosh (loga:), (vi) (sechx)/a:, (vii) sinh (sin a;), (viii) t a n - 1 (sinh a:), (ix) log tanh 2 a:, (x) exp(sinh x + cosh ar), (xi) cosec -1 (cosha:), (xii) cosh 2 x — sinh 2 x.