solve each equation for 0 less than or equal to x greater than 2ð.
cot²x-cscx=1
thanks:)
step by step please :)
cot²x-cscx=1
Use 1+cot²(x)=csc²(x)
(csc²(x)-1-csc(x)-1=0
csc²(x)-csc(x)-2=0
Let c=csc(x)
c²-c-2=0
Solve for c:
(c-2)(c+1)=0
c=2 or c=-1
=> sin(x)=1/2 or sin(x)=-1
=> x=π/6, 5π/6 or x=3π/2
The solution can be obtained by memorized exact values of sin(x), or you can see plot of sin(x) between 0 to 2π for solution.
http://img846.imageshack.us/i/1299722695.png/
To solve the equation cot²x - cscx = 1, we will follow these steps:
Step 1: Rewrite the equation using trigonometric identities.
Since cot²x = 1 + csc²x, we can substitute this identity into the equation:
(1 + csc²x) - cscx = 1
Step 2: Simplify the equation.
Distribute the negative sign to get:
1 + csc²x - cscx = 1
Combine like terms:
csc²x - cscx = 0
Step 3: Factor out a common term.
Since both terms contain cscx, we can factor it out:
cscx(cscx - 1) = 0
Step 4: Set each factor equal to zero.
To find the solutions, we set each factor equal to zero:
cscx = 0 or cscx - 1 = 0
Step 5: Solve for x in each equation.
For cscx = 0, we find the values of x where cscx equals zero. Remember that cscx represents the reciprocal of sinx:
sinx = 1 / cscx
Since cscx = 0, sinx cannot equal 0 because sinx divided by 0 is undefined. However, cscx can equal zero, which means sinx will be undefined. Therefore, there are no solutions for this equation.
For cscx - 1 = 0, we solve for x:
cscx - 1 = 0
cscx = 1
sinx = 1
x = π/2 + 2πn, where n is an integer value.
In conclusion, the solution to the equation cot²x - cscx = 1, within the given range, is x = π/2 + 2πn, where n is an integer value.