Question

(tga+ctga)*(1-cos4a)=4sin2a

Answer 1

Ответ во вложении) желаю удачи))

Answer 2

To verify the given trigonometric identity (tga+ctga)*(1-cos4a) = 4sin2a, we'll simplify both sides step by step:

1. Rewrite ctga as cot(a) (cotangent is the reciprocal of tangent).
2. Expand the left-hand side (LHS) using the distributive property.
3. Apply trigonometric identities to simplify the expression.

Starting with the left-hand side (LHS):

LHS = (tga + cot(a)) * (1 - cos(4a))

Now, we'll use the following trigonometric identities:

1. tan(a) = sin(a) / cos(a)
2. cot(a) = 1 / tan(a) = cos(a) / sin(a)
3. cos(2a) = 1 - 2sin^2(a)
4. cos(4a) = 1 - 2sin^2(2a)

Step 1: Substitute ctga with cot(a):

LHS = (tan(a) + cot(a)) * (1 - cos(4a))

Step 2: Expand using the distributive property:

LHS = tan(a) - tan(a)cos(4a) + cot(a) - cot(a)cos(4a)

Step 3: Substitute cot(a) with cos(a)/sin(a) and apply the double angle formula for cosine:

LHS = sin(a)/cos(a) - sin(a)/cos(a) * (1 - 2sin^2(2a)) + cos(a)/sin(a) - cos(a)/sin(a) * (1 - 2sin^2(2a))

Step 4: Find a common denominator (cos(a)sin(a)) for both fractions:

LHS = (sin(a)sin(a) - sin(a)sin(a)(1 - 2sin^2(2a)) + cos(a)cos(a) - cos(a)cos(a)(1 - 2sin^2(2a))) / (cos(a)sin(a))

Step 5: Simplify the terms:

LHS = (sin^2(a) + 2sin^2(2a)sin^2(a) + cos^2(a) + 2sin^2(2a)cos^2(a)) / (cos(a)sin(a))

Step 6: Apply the identity sin^2(x) + cos^2(x) = 1:

LHS = (1 + 2sin^2(2a)sin^2(a) + 2sin^2(2a)cos^2(a)) / (cos(a)sin(a))

Step 7: Apply the double angle formula for sine:

sin(2x) = 2sin(x)cos(x)

sin^2(2a) = (2sin(a)cos(a))^2 = 4sin^2(a)cos^2(a)

LHS = (1 + 2 * 4sin^2(a)cos^2(a)sin^2(a) + 2sin^2(2a)cos^2(a)) / (cos(a)sin(a))

Step 8: Combine terms:

LHS = (1 + 8sin^4(a)cos^2(a) + 2sin^2(2a)cos^2(a)) / (cos(a)sin(a))

Step 9: Apply the double angle formula for sine again:

sin(2x) = 2sin(x)cos(x)

sin^2(2a) = (2sin(a)cos(a))^2 = 4sin^2(a)cos^2(a)

LHS = (1 + 8sin^4(a)cos^2(a) + 2 * 4sin^2(a)cos^2(a) * cos^2(a)) / (cos(a)sin(a))

Step 10: Simplify further:

LHS = (1 + 8sin^4(a)cos^2(a) + 8sin^2(a)cos^4(a)) / (cos(a)sin(a))

Step 11: Apply the Pythagorean identity sin^2(x) + cos^2(x) = 1:

LHS = (1 + 8sin^4(a)(1 - sin^2(a)) + 8sin^2(a)(1 - sin^2(a))) / (cos(a)sin(a))

Step 12: Simplify again:

LHS = (1 + 8sin^4(a) - 8sin^6(a) + 8sin^2(a) - 8sin^4(a)) / (cos(a)sin(a))

Step 13: Combine like terms:

LHS = (1 + 8sin^2(a) - 8sin^6(a)) / (cos(a)sin(a))

Step 14: Apply the trigonometric identity sin^2(x) + cos^2(x) = 1:

LHS = (1 + 8sin^2(a) - 8sin^6(a)) / (cos(a)sin(a))

Step 15: Apply the double angle formula for sine:

sin(2x) = 2sin(x)cos(x)

2sin^2(a)cos(a) = sin(2a)

LHS = (1 + 8sin^2(a) - 8sin^6(a)) / (sin(2a))

Step 16: Apply the identity sin^2(x) = 1 - cos^2(x):

sin^2(a) = 1 - cos^2(a)

LHS = (1 + 8(1 - cos^2(a)) - 8(1 - cos^2(a))^3) / (sin(2a))

Step 17: Expand the terms:

LHS = (1 + 8 - 8cos^2(a) - 8(1 - 3cos^2(a) + 3cos^4(a) - cos^6(a))) / (sin(2a))

Step 18: Combine like terms:

LHS = (9 - 8cos^2(a) - 24cos^2(a) + 24cos^4(a) - 8cos^6(a)) / (sin(2a))

Step 19: Simplify further:

LHS = (9 - 32cos^2(a) + 24cos^4(a) - 8cos^6(a)) / (sin(2a))

Now, the left-hand side (LHS) is simplified. Let's simplify the right-hand side (RHS) of the identity, which is 4sin^2(a):

RHS = 4sin^2(a)

The given identity is tga+ctga)*(1-cos4a) = 4sin2a

Comparing the LHS and RHS, we see that they are not equal. Therefore, the given trigonometric identity is not valid. There might be an error or a mistake in the original expression provided. Please recheck the identity, or let me know if you have any other questions!