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Simplify $\cos ec\left( x \right) \times (\sin x + \cos x)$.

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Last updated date: 15th Jul 2024
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Answer
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Hint: The given question deals with basic simplification of trigonometric functions by using some of the simple trigonometric formulae such as $\cos ec(x) = \dfrac{1}{{\sin (x)}}$ and $\cot (x) = \dfrac{{\cos (x)}}{{\sin (x)}}$. Basic algebraic rules and trigonometric identities are to be kept in mind while doing simplification in the given problem.

Complete step-by-step solution:
In the given problem, we have to simplify the product of $\cos ec(x)$ and $[\sin (x) + \cos (x)]$.
So, $\cos ec(x) \times (\sin x + \cos x)$
Using $\cos ec(x) = \dfrac{1}{{\sin (x)}}$,
$ = $ $\dfrac{1}{{\sin x}} \times \left( {\sin x + \cos x} \right)$
On opening brackets and simplifying, the denominator $\sin (x)$ gets distributed to both the terms. So, we get,
$ = $ $\dfrac{{\sin x}}{{\sin x}} + \dfrac{{\cos x}}{{\sin x}}$
Now, cancelling the numerator and denominator in the first term, we get,
$ = $$\left( {1 + \dfrac{{\cos (x)}}{{\sin (x)}}} \right)$
Now, using $\cot (x) = \dfrac{{\cos (x)}}{{\sin (x)}}$ , we get,
$ = $$(1 + \cot x)$
Hence, the product $\cos ec\left( x \right) \times (\sin x + \cos x)$can be simplified as $(1 + \cot x)$ by the use of basic algebraic rules and simple trigonometric formulae.

Note: Trigonometric functions are also called Circular functions. Trigonometric functions are the functions that relate an angle of a right angled triangle to the ratio of two side lengths. There are $6$ trigonometric functions, namely: $\sin (x)$,$\cos (x)$,$\tan (x)$,$\cos ec(x)$,$\sec (x)$and \[\cot \left( x \right)\] . Also, $\cos ec(x)$ ,$\sec (x)$and \[\cot \left( x \right)\] are the reciprocals of $\sin (x)$,$\cos (x)$ and $\tan (x)$ respectively.