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Question

Question: How do you simplify \[\sin \theta +\cot \theta \cos \theta \]?...

How do you simplify sin⁡θ+cot⁡θcos⁡θ\sin \theta +\cot \theta \cos \theta ?

Explanation

Solution

From the question given, we have been asked to simplify sin⁡θ+cot⁡θcos⁡θ\sin \theta +\cot \theta \cos \theta . To solve the given, we have to use the basic formulae of trigonometry. After using the basic formulae of trigonometry to the given question, we have to simplify further to get the final accurate and exact answer.

Complete step by step answer:
From the question, we have been given that sin⁡θ+cot⁡θcos⁡θ\sin \theta +\cot \theta \cos \theta
From the basic formulae of trigonometry, we already know that cot⁡θ=cos⁡θsin⁡θ\cot \theta =\dfrac{\cos \theta }{\sin \theta }
Now, we have to substitute the above formula in the given question.
By substituting the above formula in the given question, we get
sin⁡θ+cot⁡θcos⁡θ\sin \theta +\cot \theta \cos \theta
⇒sin⁡θ+(cos⁡θsin⁡θ)cos⁡θ\Rightarrow \sin \theta +\left( \dfrac{\cos \theta }{\sin \theta } \right)\cos \theta
Now, as we have already discussed earlier, we have to simplify further to get the exact answer for the given question.
By simplifying the above obtained trigonometric expression further, we get
⇒sin⁡θ+cos⁡2θsin⁡θ\Rightarrow \sin \theta +\dfrac{{{\cos }^{2}}\theta }{\sin \theta }
⇒sin⁡2θ+cos⁡2θsin⁡θ\Rightarrow \dfrac{{{\sin }^{2}}\theta +{{\cos }^{2}}\theta }{\sin \theta }
From the general identities of trigonometry, we already know that sin⁡2θ+cos⁡2θ=1{{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1
Now, we have to substitute the value of the above identity in the above trigonometric expression to get the final answer.
By substituting the value of the above identity in the above trigonometric expression, we get ⇒1sin⁡θ\Rightarrow \dfrac{1}{\sin \theta }
From the basic formulae of trigonometry, we already know that 1sin⁡θ=csc⁡θ\dfrac{1}{\sin \theta }=\csc \theta
Therefore sin⁡θ+cot⁡θcos⁡θ=csc⁡θ\sin \theta +\cot \theta \cos \theta =\csc \theta
Hence, the given question is simplified by using the basic formulae of trigonometry and general identity of trigonometry.

Note:
We should be well aware of the basic formulae of trigonometry and also be well aware of the general identities of the trigonometry. We should be very careful while doing the calculation part for the given question. We should know what formula is to be used to solve the given question. Similar to the trigonometric identity sin⁡2θ+cos⁡2θ=1{{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1 we used above we also have 2 more identities they are 1+tan⁡2θ=sec⁡2θ1+{{\tan }^{2}}\theta ={{\sec }^{2}}\theta and 1+cot⁡2θ=csc⁡2θ1+{{\cot }^{2}}\theta ={{\csc }^{2}}\theta .