1+sin2x/sin2x=1+1/2(secx)(cscx)
1 + sin2x = 1 + 2sinx cosx
divide by 2sinx cosx to get
1/(2sinx cosx) + 1
= 1/2 cscx secx + 1
To prove that the equation 1 + sin(2x) / sin(2x) = 1 + 1/2(sec(x))(csc(x)) is true, we will simplify both sides of the equation step by step.
Starting with the left-hand side (LHS):
1 + sin(2x) / sin(2x)
To simplify the above expression, we note that the term sin(2x) / sin(2x) is equal to 1 since any number divided by itself is always 1.
Therefore, the LHS simplifies to:
1 + 1 = 2
Moving on to the right-hand side (RHS):
1 + 1/2(sec(x))(csc(x))
To simplify the RHS, let's recall the definitions of the trigonometric reciprocal functions:
- sec(x) = 1 / cos(x)
- csc(x) = 1 / sin(x)
Substituting these definitions into the equation:
1 + 1/2 (1/cos(x)) (1/sin(x))
Next, let's combine the fractions by taking the least common denominator. The least common denominator here is 2 * cos(x) * sin(x). So, we multiply the numerator and denominator of the fraction by 2 * cos(x) * sin(x):
(2 * cos(x) * sin(x) + 1) / (2 * cos(x) * sin(x))
Now, let's simplify the numerator further:
1 + 2 * cos(x) * sin(x) / (2 * cos(x) * sin(x))
We can simplify the above expression by dividing the numerator and denominator by 2 * cos(x) * sin(x):
1/1 = 1
So, the RHS simplifies to:
1
Since the LHS and RHS both simplify to the same value, which is 2 = 1, the original equation 1 + sin(2x) / sin(2x) = 1 + 1/2(sec(x))(csc(x)) is not true.
To solve the equation: 1 + sin(2x) / sin(2x) = 1 + 1 / 2(sec(x))(csc(x)), we can simplify the expressions on both sides of the equation.
Step 1: Simplify the left side of the equation.
The term sin(2x) / sin(2x) can be simplified to 1 since any value divided by itself is equal to 1.
Therefore, the left side of the equation simplifies to: 1 + 1 = 2.
Step 2: Simplify the right side of the equation.
The term 1 / 2(sec(x))(csc(x)) can be simplified as follows:
- sec(x) can be rewritten as 1 / cos(x).
- csc(x) can be rewritten as 1 / sin(x).
Therefore, the right side of the equation can be simplified to: 1 / (2 * (1 / cos(x)) * (1 / sin(x))).
Step 3: Simplify the expression on the right side further.
Multiplying 2 * (1 / cos(x)) * (1 / sin(x)), we get: 2 / (cos(x) * sin(x)).
Therefore, the right side of the equation simplifies to: 2 / (cos(x) * sin(x)).
Step 4: Finalize the equation.
After simplification, the equation becomes: 2 = 2 / (cos(x) * sin(x)).
At this point, the equation is simplified and doesn't have any obvious solutions, as both sides cannot be equal. Please re-check the given equation or provide additional information if needed.