How do I verify this? Cos^4x-Sin^4x=1-tan^2x/sec^2x
To verify the given equation cos^4x - sin^4x = 1 - tan^2x/sec^2x, we can simplify both sides separately and show that they are equivalent.
Step 1: Recall the trigonometric identity cos^2x - sin^2x = 1.
Step 2: Rewrite the equation using this identity:
cos^4x - sin^4x = (cos^2x + sin^2x)(cos^2x - sin^2x) = 1(cos^2x - sin^2x).
Step 3: Substitute tan^2x = sin^2x/cos^2x and sec^2x = 1/cos^2x into the right side of the equation:
1 - tan^2x/sec^2x = 1 - (sin^2x/cos^2x) / (1/cos^2x) = 1 - sin^2x = cos^2x.
Step 4: Rewrite the equation using cos^2x instead of 1:
cos^4x - sin^4x = cos^2x.
Step 5: Expand the left side of the equation using the binomial expansion formula:
(cos^2x)^2 - (sin^2x)^2 = cos^2x - sin^2x = cos^2x.
Step 6: Since both sides of the equation simplify to cos^2x, we have verified that cos^4x - sin^4x = 1 - tan^2x/sec^2x.
To verify the equation cos^4x - sin^4x = 1 - tan^2x/sec^2x, we can simplify both sides and show that they are equal to each other.
Let's start by simplifying the left side of the equation:
cos^4x - sin^4x
Using the identity a^2 - b^2 = (a + b)(a - b), we can rewrite this expression as:
(cos^2x + sin^2x)(cos^2x - sin^2x)
Since cos^2x + sin^2x = 1 (by the Pythagorean identity), we can substitute this value in:
(1)(cos^2x - sin^2x)
Now, using the identity cos^2x - sin^2x = cos(2x), we get:
cos(2x)
Now, let's simplify the right side of the equation:
1 - tan^2x/sec^2x
Using the identity sec^2x = 1 + tan^2x, we can substitute this value in:
1 - tan^2x/(1 + tan^2x)
Now, let's simplify by multiplying the numerator and denominator by (1 + tan^2x):
(1 - tan^2x)(1 + tan^2x)/(1 + tan^2x)
Using the identity 1 - tan^2x = 1/sec^2x, we can simplify further:
1/sec^2x
Since sec^2x = 1/cos^2x, we can substitute this value in:
1/(1/cos^2x)
Now, by multiplying the numerator and denominator by cos^2x, we get:
cos^2x
Therefore, the right side simplifies to cos^2x.
Now, we can see that the left side of the equation (cos(2x)) is equal to the right side (cos^2x). Thus, we have verified the equation cos^4x - sin^4x = 1 - tan^2x/sec^2x.
well, you could start out with a little algebra:
cos^4-sin^4 = (cos^2-sin^2)(cos^2+sin^2)
so you have
cos^2-sin^2
...