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Q.

The expression tan(90−θ)   is equal to _______.


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a

cosθ  

b

sinθ  

c

sinθ+cosθ  

d

cotθ   

answer is D.

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Detailed Solution

According to the question, we are given a trigonometric expression and we have to solve it. Also, we are given four Options and we have to pick the correct Options after solving it.
The given expression = tan(90−θ)  ………………………(1)
We can observe that all the given four Options in terms of θ while the above expression is in terms of (90−θ).
So, the above expression needs to be simplified more.
We know that tanθ   is the ratio of sinθ and cosθ,tanθ= sinθ cosθ  …………………..(2)
Now, on replacing θ by (90−θ)   in equation (2), we get
tan(90−θ)= sin(90−θ) cos(90−θ)  ……………………(3)
Using equation (3) and on modifying equation (1), we get
The given expression =tan(90−θ)= sin(90−θ) cos(90−θ)  ………………..(4)
We know the identity that sin(90−θ)=cosθ  …………………..(5)
We also know the identity that cos(90−θ)=sinθ  ………………….(6)
Now, from equation (4), equation (5), and equation (6), we get
tan(90−θ)= sin(90−θ) cos(90−θ) = cosθ sinθ  …………………….(7)
We know that cotθ  is the ratio of cosθ   and sinθ,cotθ= cosθ sinθ  …………………….(8)
Now, from equation (7) and equation (8), we get
tan(90−θ)= sin(90−θ) cos(90−θ) = cosθ sinθ =cotθ  ………………..(9)
Therefore, the value of the given expression tan(90−θ)   is equal to cotθ  .
 
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