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Inverse Trigonometric Functions

Short Answer Type

41.

Prove the following :

$3 space sin to the power of negative 1 end exponent straight x equals sin to the power of negative 1 end exponent left parenthesis 3 straight x minus 4 straight x cubed right parenthesis comma space straight x element of open square brackets negative 1 half comma 1 half close square brackets$

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42.

Prove the following :

$3 space cos to the power of negative 1 end exponent straight x equals cos to the power of negative 1 end exponent thin space left parenthesis 4 straight x cubed minus 3 straight x right parenthesis comma space straight x element of open square brackets 1 half comma space 1 close square brackets$

143 Views

43.

Prove space that space fraction numerator 9 straight pi over denominator 8 end fraction minus 9 over 4 space sin to the power of negative 1 end exponent 1 third equals 9 over 4 sin to the power of negative 1 end exponent fraction numerator 2 square root of 2 over denominator 3 end fraction.

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44.

Show that $tan to the power of negative 1 end exponent 1 half plus tan to the power of negative 1 end exponent 2 over 11 equals tan to the power of negative 1 end exponent 3 over 4.$

107 Views

45.

Prove that $sin to the power of negative 1 end exponent space straight x plus space cos to the power of negative 1 end exponent space straight x equals straight pi over 2$

170 Views

46.

Prove that $tan to the power of negative 1 end exponent straight x plus cot to the power of negative 1 end exponent straight x equals straight pi over 2 comma space straight x element of straight R$

116 Views

47.

Prove that $cos e c to the power of negative 1 end exponent x plus s e c to the power of negative 1 end exponent x equals straight pi over 2 comma space open vertical bar x close vertical bar greater or equal than 1$

99 Views

48. Prove that $sin to the power of negative 1 end exponent x plus sin to the power of negative 1 end exponent x y equals sin to the power of negative 1 end exponent open square brackets x square root of 1 minus y squared end root plus square root of 1 minus x squared end root close square brackets$

Put sin^–1.x = θ, sin^–1 y = ϕ
∴x = sin θ, y = sin ϕ
Now sin (θ + ϕ) = sin θ cos ϕ + cos θ sinϕ

$therefore space space sin left parenthesis straight theta plus straight ϕ right parenthesis sinθ square root of 1 minus sin squared straight ϕ end root plus square root of 1 minus sin squared straight theta. sinϕ end root rightwards double arrow space space space straight theta plus straight ϕ equals sin to the power of negative 1 end exponent open square brackets sinθ square root of 1 minus sin squared straight ϕ end root plus sinϕ square root of 1 minus sin squared straight theta end root close square brackets rightwards double arrow space space space sin to the power of negative 1 end exponent straight x plus sin to the power of negative 1 end exponent straight y equals sin to the power of negative 1 end exponent open square brackets straight x square root of 1 minus straight y squared end root plus straight y square root of 1 minus straight x squared end root close square brackets$