Question

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

Answer 1

Step-by-step explanation:

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)

as the value is muliplied by 1/2 so we get 1/2 * 2/sin (x)

= 1/sin x

 

So, the two sides are equal.

Answer 2