Inverse Trigonometric Functions

Question

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

Solution

Let $tan to the power of negative 1 end exponent straight x equals straight theta space so space that space minus straight pi over 2 less than straight theta less than straight pi over 2$
$rightwards double arrow straight pi over 2 greater than negative straight theta greater than negative straight pi over 2 space space space space space space space space space space rightwards double arrow straight pi over 2 plus straight pi over 2 greater than straight pi over 2 minus straight theta greater than straight pi over 2 minus straight pi over 2 rightwards double arrow space straight pi greater than straight pi over 2 minus straight theta greater than 0 space space space space space space space space space space space space space rightwards double arrow space straight pi over 2 minus straight theta element of left parenthesis 0 comma straight pi right parenthesis because space space space space tan to the power of negative 1 end exponent straight x equals straight theta space space space space space space space space space space space space space space space space therefore space space space space space straight x equals tan space straight theta space space space space space space rightwards double arrow space space space straight x equals cot open parentheses straight pi over 2 minus straight theta close parentheses. space But space straight x over 2 minus straight theta element of left parenthesis 0 comma straight pi right parenthesis therefore space space space space space cot to the power of negative 1 end exponent straight x equals straight pi over 2 minus straight theta rightwards double arrow space space space space cos to the power of negative 1 end exponent straight x equals straight pi over 2 minus tan to the power of negative 1 end exponent straight x space space space space space space rightwards double arrow space tan to the power of negative 1 end exponent straight x plus cot to the power of negative 1 end exponent straight x equals straight pi over 2 space for all space straight x$

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Inverse Trigonometric Functions

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

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