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Question: How do you use the fundamental identities to simplify \(\cot \theta \sec \theta \) ?...

How do you use the fundamental identities to simplify cot⁡θsec⁡θ\cot \theta \sec \theta ?

Explanation

Solution

In this question, we have to find the value of trigonometric function. Thus, we will use the fundamental identities to get the solution. First, we will change the cot function and secant function in terms of sine and cosine function using the formula cot⁡θ=cos⁡θsin⁡θ\cot \theta =\dfrac{\cos \theta }{\sin \theta } and sec⁡θ=1cos⁡θ\sec \theta =\dfrac{1}{\cos \theta } . After that, we will put these values in the given trigonometric function. In the last, we will cancel out the same terms in the division and again apply the trigonometric formula sin⁡θ=1csc⁡θ\sin \theta =\dfrac{1}{\csc \theta } , to get the required solution of the problem.

Complete step by step answer:
According to the problem, we have to find the value of the trigonometric function.
Thus, we will apply the fundamental identities to get the solution.
The trigonometric function given to us is cot⁡θsec⁡θ\cot \theta \sec \theta ---------- (1)
Now, we will first change the given functions in terms of sine and cosine function, that is we will apply the trigonometric identity cot⁡θ=cos⁡θsin⁡θ\cot \theta =\dfrac{\cos \theta }{\sin \theta } and sec⁡θ=1cos⁡θ\sec \theta =\dfrac{1}{\cos \theta } in equation (1), we get
⇒cos⁡θsin⁡θ×1cos⁡θ\Rightarrow \dfrac{\cos \theta }{\sin \theta }\times \dfrac{1}{\cos \theta }
As we know, the same terms in the division will cancel out each other with the remainder 0 and quotient 1, thus we get
⇒1sin⁡θ\Rightarrow \dfrac{1}{\sin \theta }
In the last, we will again apply the trigonometric formula sin⁡θ=1csc⁡θ\sin \theta =\dfrac{1}{\csc \theta } in the above expression, we get
⇒11csc⁡θ\Rightarrow \dfrac{1}{\dfrac{1}{\csc \theta }}
Now, we will take the reciprocal of the function in the denominator, thus we get
⇒csc⁡θ\Rightarrow \csc \theta which is the required solution.
Therefore, using the fundamental identities to simplify the trigonometric function cot⁡θsec⁡θ\cot \theta \sec \theta , we get the value equal to csc⁡θ\csc \theta .

Note:
While solving this problem, do mention all the steps carefully and avoid confusion and mathematical error. Always write the angle θ\theta instead of any other algebraic variable. You can also stop your solution after this 1sin⁡θ\dfrac{1}{\sin \theta } step.