Prove the following-sin -1 x-sin -1 y-sin -1 z- Show that x -1-x2-y -1-y2-z -1-z2-x y z

Sin^ 1x+sin^ 1y+sin^ 1z=π Let x=sina, y=sin b and z=sinc ∴ a+b+c=π x√(1 x^2)+y√(1 y^2)+z√(1 z^2) =sin a √(1 sin^2 a)+sin b√(1 sin^2 b)+sin c√(1 sin^2 c) =si ...

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Byju's Answer

Standard XII

Mathematics

Property 1

Prove the fol...

Question

Prove the following:
$s i n^{- 1} x + s i n^{- 1} y + s i n^{- 1} z = π$ Show that $x \sqrt{1 - x^{2}} + y \sqrt{1 - y^{2}} + z \sqrt{1 - z^{2}} = x y z$

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Solution

$s i n^{- 1} x + s i n^{- 1} y + s i n^{- 1} z = π L e t x = s i n a, y = s i n b a n d z = s i n c ∴ a + b + c = π x \sqrt{1 - x^{2}} + y \sqrt{1 - y^{2}} + z \sqrt{1 - z^{2}} = s i n a \sqrt{1 - s i n^{2} a} + s i n b \sqrt{1 - s i n^{2} b} + s i n c \sqrt{1 - s i n^{2} c} = s i n a c o s a + s i n b c o s b + s i n c c o s c = \frac{s i n 2 a + s i n 2 b + s i n 2 c}{2} = \frac{2 s i n \frac{2 a + 2 b}{2} c o s \frac{2 a - 2 b}{2} + s i n 2 c}{2} = \frac{2 s i n a + b c o s a - b s i n 2 c}{2} = \frac{2 s i n (π - c) c o s (a - b) + 2 s i n c c o s c}{2} = \frac{2 s i n c c o s (a - b) + 2 s i n c c o s c}{2} = s i n c [c o s (a - b) + c o s c] = s i n c [2 c o s (\frac{a - b + c}{2}) c o s (\frac{a - b - c}{2})] = s i n c [2 c o s \frac{π - b - b}{2} c o s (\frac{a - π + a}{2})] = s i n c [2 c o s (\frac{π}{2} - b) c o s (a - \frac{π}{2})] = s i n c \times 2 \times s i n b \times s i n a = 2 s i n a s i n b s i n c = 2 x y z$

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Similar questions

Q. If

{sin}^{- 1} x + {sin}^{- 1} y + {sin}^{- 1} z = π

, prove that

x \sqrt{1 - x^{2}} + y \sqrt{1 - y^{2}} + z \sqrt{1 - z^{2}} = 2 x y z

Inverse circular functions,Principal values of

{s i n}^{- 1} x, {c o s}^{- 1} x, {t a n}^{- 1} x

{t a n}^{- 1} x + {t a n}^{- 1} y = {t a n}^{- 1} \frac{x + y}{1 - x y}

x y < 1

π + {t a n}^{- 1} \frac{x + y}{1 - x y}

x y > 1

(a) If

{c o s}^{- 1} p + {c o s}^{- 1} q + {c o s}^{- 1} r = π

, then prove that

p^{2} + q^{2} + r^{2} + 2 p q r = 1

(b) If

{s i n}^{- 1} x + {s i n}^{- 1} y + {s i n}^{- 1} z = π

, then prove that

x^{4} + y^{4} + z^{4} + 4 x^{2} y^{2} z^{2} = 2 (x^{2} y^{2} + y^{2} z^{2} + z^{2} x^{2})

{t a n}^{- 1} x + {t a n}^{- 1} y + {t a n}^{- 1} z = π

π / 2

show that

x + y + z = x y z

x y + y z + z x = 1

Q. Prove that :

{sin}^{- 1} x + {sin}^{- 1} y = {sin}^{- 1} [x \sqrt{1 - y^{2}} + y \sqrt{1 - x^{2}}]

Q. If

{sin}^{- 1} x + {sin}^{- 1} y + {sin}^{- 1} z = π

, then prove that

x \sqrt{1 - x^{2}} + y \sqrt{1 - y^{2}} + z \sqrt{1 - z^{2}} = 2 x y z

Q. If

{cos}^{- 1} x + {cos}^{- 1} y + {cos}^{- 1} z = π

, prove that

x^{2} + y^{2} + z^{2} + 2 x y z = 1

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Prove the following: sin−1x+sin−1y+sin−1z=π Show that x√1−x2+y√1−y2+z√1−z2=xyz

Prove the following:
$s i n^{- 1} x + s i n^{- 1} y + s i n^{- 1} z = π$ Show that $x \sqrt{1 - x^{2}} + y \sqrt{1 - y^{2}} + z \sqrt{1 - z^{2}} = x y z$