Question

Evaluate \[tan\left( {x + \dfrac{\pi }{4}} \right)\]

Answer 1

Hint: We have to evaluate the value of \[tan\left( {x + \dfrac{\pi }{4}} \right)\] . We solve the question by using trigonometric identities and the values of trigonometric functions . We use the formula of tan of sum of two angles and after expanding the formula and putting the values we get the value of \[tan\left( {x + \dfrac{\pi }{4}} \right)\] .

Complete step-by-step answer:
All the trigonometric functions are classified into two categories or types as either sine function or cosine function . All the functions which lie in the category of sine functions are sin , cosec and tan functions on the other hand the functions which lie in the category of cosine functions are cos , sec and cot functions . The trigonometric functions are classified into these two categories on the basis of their property which is stated as : when the value of angle is substituted by the negative value of the angle then we get the negative value for the functions in the sine family and a positive value for the functions in the cosine family .
Given : To evaluate \[tan\left( {x + \dfrac{\pi }{4}} \right)\]
Using the formula of \[tan\left( {a + b} \right) = \dfrac{{\left[ {tana + tanb} \right]}}{{\left[ {1 - tana \times tanb} \right]}}\]
Expanding \[tan\left( {x + \dfrac{\pi }{4}} \right)\] using the above formula , we get \[tan\left( {x + \dfrac{\pi }{4}} \right)\] \[ = \dfrac{{[tanx + tan\dfrac{\pi }{4}}}{{{\text{ [}}1 - tanx \times tan\dfrac{\pi }{4}]}}\]
As , \[tan\dfrac{\pi }{4} = 1\] and putting in the equation
\[tan\left( {x + \dfrac{\pi }{4}} \right) = \dfrac{{\left[ {tanx + 1} \right]}}{{\left[ {1 - tanx} \right]}}\]
Hence , the value of \[tan\left( {x + \dfrac{\pi }{4}} \right) = \dfrac{{\left[ {1 + tanx} \right]}}{{\left[ {1 - tanx} \right]}}\]
So, the correct answer is “Option B”.

Note: We have various trigonometric formulas used to solve the problem
The various trigonometric formulas used :
\[sin\left( {a + b} \right) = sina \times cosb + sinb \times cosa\]
\[sin\left( {a - b} \right) = sina \times cosb - sinb \times cosa\]
\[cos\left( {a + b} \right) = cosa \times cosb - sinb \times sina\]
\[cos\left( {a - b} \right) = cosa \times cosb + sinb \times sina\]
All the trigonometric functions are positive in first quadrant , the sin function are positive in second quadrant and rest are negative , the tan function are positive in third quadrant and rest are negative , the cos function are positive in fourth quadrant and rest are negative .