Someone please help me prove these

A) cos^4x-sin^4x=cos2x

B)tanx=sin^2x/1+cos^2x

Please and Thank you

looks like you need to review your basic identities and double-angle formulas:

cos^4x-sin^4x
= (cos^2x-sin^2x)(cos^2x+sin^2x)
= cos^2x-sin^2x
= cos2x

(B) is clearly not true, since sin^2x/(1+cos^2x) has no asymptotes

Sure, I can help you prove both of these trigonometric identities.

A) To prove cos^4(x) - sin^4(x) = cos(2x), we will use the double-angle identity for cosine. The double-angle identity for cosine states that cos(2x) = cos^2(x) - sin^2(x).

Step 1: Start with the left-hand side: cos^4(x) - sin^4(x).
Step 2: Use the identity cos^4(x) = (cos^2(x))^2 and sin^4(x) = (sin^2(x))^2.
Therefore, the equation can be rewritten as (cos^2(x))^2 - (sin^2(x))^2.

Step 3: Apply the identity a^2 - b^2 = (a + b)(a - b).
The equation becomes (cos^2(x) + sin^2(x))(cos^2(x) - sin^2(x)).

Step 4: According to the Pythagorean identity, cos^2(x) + sin^2(x) = 1, so the equation simplifies to 1(cos^2(x) - sin^2(x)).

Step 5: Use the double-angle identity for cosine: cos(2x) = cos^2(x) - sin^2(x).
So, we can replace cos^2(x) - sin^2(x) with cos(2x) in the equation.

Final step: Therefore, cos^4(x) - sin^4(x) simplifies to 1 * cos(2x) = cos(2x), which proves the identity.

B) For tan(x) = sin^2(x)/(1 + cos^2(x)), we will use the trigonometric identity tan(x) = sin(x)/cos(x).

Step 1: Start with the left-hand side: tan(x) = sin^2(x)/(1 + cos^2(x)).

Step 2: Use the identity sin^2(x) = 1 - cos^2(x).

The equation becomes tan(x) = (1 - cos^2(x))/(1 + cos^2(x)).

Step 3: Combine like terms by finding a common denominator of (1 + cos^2(x)).

The equation simplifies to tan(x) = (1 - cos^2(x))/(1 + cos^2(x)) * (1 - cos^2(x))/(1 - cos^2(x)).

Step 4: Multiply the numerator and denominator together.

The equation becomes tan(x) = (1 - 2cos^2(x) + cos^4(x))/(1 - cos^4(x)).

Step 5: Use the identity cos^2(x) = 1 - sin^2(x) to replace cos^4(x) and 1 - cos^4(x).

The equation simplifies to tan(x) = (1 - 2cos^2(x) + 1 - sin^2(x))/(1 - (1 - sin^2(x))).

Step 6: Continuing to simplify, we can replace 1 - 2cos^2(x) + 1 - sin^2(x) with 2 - (2cos^2(x) + sin^2(x)).

The equation becomes tan(x) = (2 - (2cos^2(x) + sin^2(x)))/(sin^2(x)).

Step 7: Recall the trigonometric identity cos^2(x) + sin^2(x) = 1, so 2cos^2(x) + sin^2(x) = 2 - 1.

The equation further simplifies to tan(x) = (2 - 1)/(sin^2(x)).

Step 8: Simplify 2 - 1 to get the final result.

tan(x) = 1/(sin^2(x)).

Therefore, the identity tan(x) = sin^2(x)/(1 + cos^2(x)) is proven.