Question

Cos^2a-4sin^2a/2 cos^2a/2 /1-8sin^2a/2cos2a/2=1​

Answer 1

Ответ:

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Answer 2

It looks like you've provided an expression involving trigonometric functions and are asking to simplify it to 1. Let's break down the expression step by step and simplify it:

Expression: (cos^2(a) - 4sin^2(a)) / (2cos^2(a/2)) / (1 - 8sin^2(a/2)cos(a/2))

Step 1: We'll start by simplifying the numerator and denominator separately.

Numerator: cos^2(a) - 4sin^2(a) Using the trigonometric identity: cos^2(a) - sin^2(a) = cos(2a) cos(2a)

Denominator: 2cos^2(a/2) Using the half-angle formula: cos^2(a/2) = (1 + cos(a)) / 2 2 * (1 + cos(a)) / 2 1 + cos(a)

1 - 8sin^2(a/2)cos(a/2) Using the double-angle formula: sin(2a) = 2sin(a)cos(a) 8sin(a/2)cos(a/2) 4 * 2sin(a/2)cos(a/2) 4sin(a)

Step 2: Substitute the simplified numerator and denominators back into the expression:

(cos(2a)) / (1 + cos(a)) / (1 - 4sin(a))

Step 3: Combine the fractions by multiplying the first fraction by the reciprocal of the second fraction:

(cos(2a)) * (1 - 4sin(a)) / (1 + cos(a))

Step 4: Distribute cos(2a) through the numerator:

cos(2a) - 4sin(a)cos(2a) / (1 + cos(a))

Step 5: Apply the double-angle formula for cosine: cos(2a) = 2cos^2(a) - 1

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

Step 6: Distribute and simplify further:

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

Step 7: Factor out common terms in the numerator:

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

Step 8: Combine like terms:

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

Step 9: Factor out common terms again:

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

Step 10: Substitute the double-angle formula for cosine: cos(2a) = 2cos^2(a) - 1

2 * cos(2a) * (1 - 4sin(a)) + 4sin(a) - 1 / (1 + cos(a))

Step 11: Distribute and simplify:

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

Step 12: Combine terms again:

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

The final simplified expression is 1 + cos(a), which is not equal to 1. Therefore, it appears there may have been a mistake or typo in the original expression or calculation. Please double-check the expression or provide additional information if necessary.