Question

(1-sin^2a)(1+tg^2a)-sin^2a

Answer 1

Ответ:

Пошаговое объяснение:

1) 1- sin^2a = cos^2a - основное тригонометрическое тождество

1+tg^2a = 1/cos^2a тоже есть такая формула => cos^2a( 1/cos^2a) = 1,  1= 1

Answer 2

To simplify the given expression (1 - sin^2(a))(1 + tan^2(a)) - sin^2(a), we'll use trigonometric identities to make the calculations:

1. Recall the Pythagorean identity for sine and tangent: sin^2(a) + cos^2(a) = 1 tan^2(a) + 1 = sec^2(a)

2. Substitute sin^2(a) in terms of cos^2(a) using the Pythagorean identity: sin^2(a) = 1 - cos^2(a)

3. Substitute tan^2(a) in terms of sec^2(a) using the Pythagorean identity: tan^2(a) = sec^2(a) - 1

Now, let's proceed with the simplification:

(1 - sin^2(a))(1 + tan^2(a)) - sin^2(a)

Substitute sin^2(a) and tan^2(a):

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

Simplify the inner expressions:

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

Now, use the identity sec^2(a) = 1 + tan^2(a):

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

Substitute tan^2(a) in terms of sec^2(a):

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

Expand the terms:

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

The cos^2(a) terms cancel out:

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

Finally, we know that sec^2(a) = 1 + tan^2(a), so:

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

Substitute cos^2(a) in terms of sin^2(a):

= (1 - sin^2(a))(1 + tan^2(a)) - 1

And there you have the simplified expression: (1 - sin^2(a))(1 + tan^2(a)) - 1.