Question

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

Answer 1

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 answer:
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.
So, the correct answer is “$ (1 + \cot x) $”.

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.
Given problem deals with Trigonometric functions. For solving such problems, trigonometric formulae should be remembered by heart such as: $ \tan (x) = \dfrac{{\sin (x)}}{{\cos (x)}} $ and $ \cot (x) = \dfrac{{\cos (x)}}{{\sin (x)}} $ .