sin/1+cos + 1+cos/sin=2cotsec

Question

prove the following is an identity thank you

Answer 1

Like Anthony says, it's really important to make sure you write your questions clearly so we don't have to guess what you mean. When tutors have to guess, they're more likely to ignore your question or give you the wrong answer.

 

 

 

I'm going to guess that your equation is sin(x) / (1 + cos(x)) + (1 + cos(x)) / sin(x) = 2(cot(x))(sec(x)).

 

Let's first work on finding a common denominator for the left-hand side of the equation. That common denominator will be (1 + cos(x))(sin(x)). So, multiply the first fraction by sin(x)/sin(x) and the second fraction by (1 + cos(x))/(1 + cos(x)), then combine the fractions:

 

sin²(x) / (sin(x)(1 + cos(x))) + (1 + cos(x))² / (sin(x)(1 + cos(x))) = 2(cot(x))(sec(x))

(sin²(x) + (1 + cos(x))²) / (sin(x)(1 + cos(x))) = 2(cot(x))(sec(x))

(sin²(x) + 1 + cos(x) + cos(x) + cos²(x)) / (sin(x)(1 + cos(x))) = 2(cot(x))(sec(x))

(sin²(x) + 1 + 2cos(x) + cos²(x)) / (sin(x)(1 + cos(x))) = 2(cot(x))(sec(x))

 

 

 

Now, remember that sin²(x) + cos²(x) = 1 and substitute that into the left-hand side:

 

(1 + 1 + 2cos(x)) / (sin(x)(1 + cos(x))) = 2(cot(x))(sec(x))

(2 + 2cos(x)) / (sin(x)(1 + cos(x))) = 2(cot(x))(sec(x))

2(1 + cos(x)) / (sin(x)(1 + cos(x))) = 2(cot(x))(sec(x))

2/sin(x) = 2(cot(x))(sec(x))

 

 

 

It seems like we've done just about all we can to the left-hand side. So, switch to the right-hand side. Let's try to get everything in terms of sine and cosine. cot = cos/sin and sec = 1/cos, so:

 

2/sin(x) = 2(cos(x)/sin(x))(1/cos(x))

2/sin(x) = 2(cos(x)/(sin(x)cos(x))

2/sin(x) = 2(1/sin(x))

2/sin(x) = 2/sin(x)

 

So, the two sides are equal.