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Inverse Trigonometric Functions

Short Answer Type

111.

Write the value of $open parentheses 2 tan to the power of negative 1 end exponent 1 fifth close parentheses$

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112.

Find the value of the following:
$tan 1 half open square brackets sin to the power of negative 1 end exponent fraction numerator 2 straight x over denominator 1 plus straight x squared end fraction plus cos to the power of negative 1 end exponent fraction numerator 1 minus straight y squared over denominator 1 plus straight y squared end fraction close square brackets comma space open vertical bar straight x close vertical bar space less than 1 comma space space straight y greater than 0 space and space xy less than 1$

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113.

Prove that $tan to the power of negative 1 end exponent open parentheses 1 half close parentheses plus tan to the power of negative 1 end exponent open parentheses 1 fifth close parentheses plus tan to the power of negative 1 end exponent open parentheses 1 over 8 close parentheses space equals straight pi over 4$

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114.
If a line makes angles 90° and 60° respectively with the positive directions of x and y axes, find the angle which it makes with the positive direction of the z-axis.

suppose the direction cosines of the line be l,m,and n.
we know that l² + m²+n² = 1
Let the line make angle θ with the positive direction of the z-axis.
α = 90°, β = 60° γ = θ
Thus,
cos² 90 + cos²60 + cos²θ =1
$rightwards double arrow space 0 space plus space open parentheses 1 half close parentheses squared space plus space cos squared straight theta space equals space 1 rightwards double arrow space cos squared space straight theta space equals space 1 minus 1 fourth rightwards double arrow space cos squared space straight theta space equals 3 over 4 rightwards double arrow space cos space straight theta space equals space plus-or-minus fraction numerator square root of 3 over denominator 2 end fraction rightwards double arrow space straight theta space equals space 30 to the power of straight o$