Question
Question: Prove that \[\dfrac{1}{{\cos ec\theta + 1}} + \dfrac{1}{{\cos ec\theta - 1}} = 2\sec \theta \tan \th...
Prove that cosecθ+11+cosecθ−11=2secθtanθ ?
Solution
The given question deals with proving a trigonometric equality using the basic and simple trigonometric formulae and identities such as cosecx=sinx1. Basic algebraic rules and trigonometric identities are to be kept in mind while simplifying the given problem and proving the result given to us.
Complete answer:
For proving the desired result, we need to have a good grip over the basic trigonometric formulae and identities.
Now, we need to make the left and right sides of the equation equal.
L.H.S. =cosecθ+11+cosecθ−11
So, we will convert all the trigonometric functions into sine and cosine using trigonometric formulae and identities. So, using the trigonometric formula cosecx=sinx1, we get,
=sinθ1+11+sinθ1−11
Taking the LCM in the denominators, we get,
=sinθ1+sinθ1+sinθ1−sinθ1
Simplifying the expression,
=1+sinθsinθ+1−sinθsinθ
Multiplying the numerator and denominator of the first term by (1−sinθ) and that of the second term by (1+sinθ).
So, we get,
=1+sinθsinθ×(1−sinθ1−sinθ)+1−sinθsinθ×(1+sinθ1+sinθ)
Using the algebraic identity a2−b2=(a−b)(a+b),
=1−sin2θsinθ(1−sinθ)+1−sin2θsinθ(1+sinθ)
Simplifying the expression, we get,
=1−sin2θsinθ−sin2θ+1−sin2θsinθ+sin2θ
Adding the numerator directly as the denominators of both the rational expression is same.
=1−sin2θsinθ+sinθ
Now, applying the trigonometric identity sin2x+cos2x=1 in the common denominator, we get,
=cos2θ2sinθ
Now, we know that tanx=cosxsinx and secx=cosx1
=2secθtanθ
Now, R.H.S =2secθtanθ
As the left side of the equation is equal to the right side of the equation, we have,
cosecθ+11+cosecθ−11=2secθtanθ
Hence, Proved.
Note:
Given problem deals with Trigonometric functions. For solving such problems, trigonometric formulae and identities such as tanx=cosxsinx and sin2x+cos2x=1 should be used. We also need knowledge of algebraic rules and identities to simplify the expression. Definitions of the trigonometric functions such as secant secx=cosx1, cosecant cosecx=sinx1 and tangent are essential for solving the problem.