Question

1) cos²a • sin²a+cos⁴a2) cos²B (1+tg²B​)

Answer 1

Ответ:

Упростить выражения .

Применяем основные тригонометрические тождества:

           $\bf sin^2a+cos^2a=1\ \ ,\ \ \ 1+tg^2a=\dfrac{1}{cos^2a}$   .

$\bf 1)\ \ cos^2a\cdot sin^2a+cos^4a=cos^2a\cdot (\underbrace{\bf sin^2a+cos^2a}_{1})=cos^2a\cdot 1=cos^2a\\\\\\2)\ \ cos^2\beta \, (1+tg^2\beta )=cos^2\beta \cdot \dfrac{1}{cos^2\beta }=\dfrac{cos^2\beta }{cos^2\beta }=1$

Answer 2

Let's simplify the given expressions:

1. cos²a • sin²a + cos⁴a:

First, recall the trigonometric identity: sin²a + cos²a = 1

Now, we can rewrite sin²a as (1 - cos²a): (1 - cos²a) • cos²a + cos⁴a

Expand the first term: cos²a - cos⁴a + cos⁴a

Now, the cos⁴a terms cancel out: cos²a

So, the simplified form of the first expression is just cos²a.

2. cos²B (1 + tan²B):

First, recall the trigonometric identity: 1 + tan²B = sec²B

Now, we can rewrite the expression as: cos²B • sec²B

Now, recall another trigonometric identity: cos²B • sec²B = 1

So, the simplified form of the second expression is 1.