prove:
sinx/sec2x+cosx/csc2x=sin3x
LS = sinx(cos2x) + cosx(sin 2x)
RS = sin(3x) = sin(2x + x)
= (sin 2x)cosx + (cos 2x) sinx
= LS
I am using sin(A+B) = sinAcosB + cosAsinB
To prove the given equation, we need to simplify both sides of the equation and show that they are equal. Let's simplify each side step by step.
Starting with the left side:
sin(x)/sec^2(x) + cos(x)/csc^2(x)
We can rewrite sec^2(x) and csc^2(x) in terms of sin(x) and cos(x):
sec^2(x) = 1/cos^2(x) and csc^2(x) = 1/sin^2(x)
Substituting these values:
sin(x)/(1/cos^2(x)) + cos(x)/(1/sin^2(x))
Next, multiply each term by the reciprocal of the denominator:
sin(x) * cos^2(x) + cos(x) * sin^2(x)
Now, let's work with the right side of the equation:
sin(3x)
Using the triple-angle identity for sine:
3sin(x) - 4sin^3(x)
Now, we need to show that these two expressions are equal:
sin(x) * cos^2(x) + cos(x) * sin^2(x) = 3sin(x) - 4sin^3(x)
To simplify further, we will use the identity sin^2(x) + cos^2(x) = 1.
Rearranging the equation, we get:
sin(x) * (1 - sin^2(x)) + cos(x) * sin^2(x) = 3sin(x) - 4sin^3(x)
Expanding the terms:
sin(x) - sin^3(x) + cos(x) * sin^2(x) = 3sin(x) - 4sin^3(x)
Now, let's combine like terms and simplify:
-sin^3(x) + cos(x) * sin^2(x) = 2sin(x) - 4sin^3(x)
The next step involves factoring out sin(x):
sin^2(x) * (-sin(x) + cos(x)) = sin(x) * (2 - 4sin^2(x))
Dividing both sides by sin(x):
sin(x) * (-sin(x) + cos(x)) = 2 - 4sin^2(x)
Now, let's use the identity sin(x) = 1 - cos(x) to rewrite (-sin(x) + cos(x)):
sin(x) * (1 - cos(x)) = 2 - 4sin^2(x)
Expanding the left side:
sin(x) - sin(x) * cos(x) = 2 - 4sin^2(x)
Using the identity sin^2(x) = 1 - cos^2(x):
sin(x) - sin(x) * cos(x) = 2 - 4(1 - cos^2(x))
Expanding further:
sin(x) - sin(x) * cos(x) = 2 - 4 + 4cos^2(x)
Combining like terms:
sin(x) - sin(x) * cos(x) = 4cos^2(x) - 2
Now, let's rearrange the terms and simplify:
sin(x) * (1 - cos(x)) = 4cos^2(x) - 2
Using the identity sin(x) = 1 - cos(x) again:
(1 - cos(x)) * (1 - cos(x)) = 4cos^2(x) - 2
Expanding both sides:
1 - 2cos(x) + cos^2(x) = 4cos^2(x) - 2
Now, let's simplify this equation:
1 - 2cos(x) + cos^2(x) = 4cos^2(x) - 2
Moving all terms to one side:
cos^2(x) - 4cos^2(x) + 2cos(x) + 1 - 2 = 0
Combining like terms:
-3cos^2(x) + 2cos(x) - 1 = 0
Now, factoring the quadratic equation:
(-cos(x) + 1)(3cos(x) - 1) = 0
Setting each factor equal to zero:
-cos(x) + 1 = 0 or 3cos(x) - 1 = 0
Solving the first equation:
-cos(x) = -1
cos(x) = 1
Since the cosine of any angle cannot exceed 1, there is no solution for cos(x) = 1.
Now, solving the second equation:
3cos(x) = 1
cos(x) = 1/3
Using the trigonometric identity sin^2(x) + cos^2(x) = 1, we can find sin(x):
sin^2(x) + (1/3)^2 = 1
sin^2(x) + 1/9 = 1
sin^2(x) = 8/9
Taking the square root of both sides, considering the positive value since sin(x) is positive in the given equation:
sin(x) = √(8/9) = ±(2√2/3)
Since the equation doesn't specify any restrictions, both positive and negative values of sin(x) are valid solutions.
Therefore, we have proven that sin(x)/sec^2(x) + cos(x)/csc^2(x) is equal to sin(3x).