Question
Question: Simplify \( \cos ec\left( x \right) \times (\sin x + \cos x) \) ....
Simplify cosec(x)×(sinx+cosx) .
Solution
Hint : The given question deals with basic simplification of trigonometric functions by using some of the simple trigonometric formulae such as cosec(x)=sin(x)1 and cot(x)=sin(x)cos(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) and [sin(x)+cos(x)] .
So, cosec(x)×(sinx+cosx)
Using cosec(x)=sin(x)1 ,
= sinx1×(sinx+cosx)
On opening brackets and simplifying, the denominator sin(x) gets distributed to both the terms. So, we get,
= sinxsinx+sinxcosx
Now, cancelling the numerator and denominator in the first term, we get,
= (1+sin(x)cos(x))
Now, using cot(x)=sin(x)cos(x) , we get,
= (1+cotx)
Hence, the product cosec(x)×(sinx+cosx) can be simplified as (1+cotx) by the use of basic algebraic rules and simple trigonometric formulae.
So, the correct answer is “(1+cotx)”.
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) , cosec(x) , sec(x) and cot(x) . Also, cosec(x) , sec(x) and cot(x)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)=cos(x)sin(x) and cot(x)=sin(x)cos(x) .