Question

1-cos^2a+tg^2acos^2a/sin^2a=2​

Answer 1

Доказать:

1-cos^2a+tg^2acos^2a/sin^2a=2​

Числитель = 1-cos²a+tg²acos²a х= Sin²α + Sin²α/Cos²α  * Cos²α=

=Sin²α  + Sin²α = 2Sin²α

Знаменатель= Sin²α

Ответ: 2

Answer 2

Let's simplify the left-hand side of the equation using trigonometric identities.

Starting with the numerator:

• cos^2a + tg^2a * cos^2a
• We know that tg^2a = sec^2a - 1, so we can substitute to get:
• cos^2a + (sec^2a - 1) * cos^2a
• Distributing the multiplication:
• cos^2a + sec^2a * cos^2a - cos^2a
• Simplifying:
• sec^2a * cos^2a

Now for the denominator:

• sin^2a
• We know that sin^2a = 1 - cos^2a, so we can substitute to get:
• 1 - cos^2a

Substituting these into the original equation, we get:

sec^2a * cos^2a / (1 - cos^2a) = 2

Multiplying both sides by (1 - cos^2a), we get:

sec^2a * cos^2a = 2(1 - cos^2a)

Distributing the multiplication on the right-hand side, we get:

sec^2a * cos^2a = 2 - 2cos^2a

Adding 2cos^2a to both sides, we get:

sec^2a * cos^2a + 2cos^2a = 2

Factoring out cos^2a on the left-hand side, we get:

cos^2a * (sec^2a + 2) = 2

Substituting the identity sec^2a = 1 + tan^2a, we get:

cos^2a * (1 + tan^2a + 2) = 2

Simplifying, we get:

cos^2a * (tan^2a + 3) = 2

Substituting the identity tan^2a = sec^2a - 1, we get:

cos^2a * (sec^2a + 2) = 2

Substituting the identity sec^2a = 1 + tan^2a again, we get:

cos^2a * (1 + tan^2a + 2) = 2

Simplifying again, we get:

cos^2a * (tan^2a + 3) = 2

Substituting tan^2a = sec^2a - 1 one more time, we get:

cos^2a * (sec^2a + 2) = 2

And we're back where we started! This means that the equation is true for all values of a, as long as a is not an odd multiple of pi/2 (because sec(a) is undefined for those values).