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Question

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

Simplify cosec(x)×(sinx+cosx)\cos ec\left( x \right) \times (\sin x + \cos x) .

Explanation

Solution

Hint : The given question deals with basic simplification of trigonometric functions by using some of the simple trigonometric formulae such as cosec(x)=1sin(x)\cos ec(x) = \dfrac{1}{{\sin (x)}} and cot(x)=cos(x)sin(x)\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 cosec(x)\cos ec(x) and [sin(x)+cos(x)][\sin (x) + \cos (x)] .
So, cosec(x)×(sinx+cosx)\cos ec(x) \times (\sin x + \cos x)
Using cosec(x)=1sin(x)\cos ec(x) = \dfrac{1}{{\sin (x)}} ,
== 1sinx×(sinx+cosx)\dfrac{1}{{\sin x}} \times \left( {\sin x + \cos x} \right)
On opening brackets and simplifying, the denominator sin(x)\sin (x) gets distributed to both the terms. So, we get,
== sinxsinx+cosxsinx\dfrac{{\sin x}}{{\sin x}} + \dfrac{{\cos x}}{{\sin x}}
Now, cancelling the numerator and denominator in the first term, we get,
== (1+cos(x)sin(x))\left( {1 + \dfrac{{\cos (x)}}{{\sin (x)}}} \right)
Now, using cot(x)=cos(x)sin(x)\cot (x) = \dfrac{{\cos (x)}}{{\sin (x)}} , we get,
== (1+cotx)(1 + \cot x)
Hence, the product cosec(x)×(sinx+cosx)\cos ec\left( x \right) \times (\sin x + \cos x) can be simplified as (1+cotx)(1 + \cot x) by the use of basic algebraic rules and simple trigonometric formulae.
So, the correct answer is “(1+cotx)(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 66 trigonometric functions, namely: sin(x)\sin (x) , cos(x)\cos (x) , tan(x)\tan (x) , cosec(x)\cos ec(x) , sec(x)\sec (x) and cot(x)\cot \left( x \right) . Also, cosec(x)\cos ec(x) , sec(x)\sec (x) and cot(x)\cot \left( x \right)are the reciprocals of sin(x)\sin (x) , cos(x)\cos (x) and tan(x)\tan (x) respectively.
Given problem deals with Trigonometric functions. For solving such problems, trigonometric formulae should be remembered by heart such as: tan(x)=sin(x)cos(x)\tan (x) = \dfrac{{\sin (x)}}{{\cos (x)}} and cot(x)=cos(x)sin(x)\cot (x) = \dfrac{{\cos (x)}}{{\sin (x)}} .