Question

Cos⁴a+sin²a×cos²a упроситите и найдите его значиние при tga=2​

Answer 1

$Cos^{4}\alpha+Sin^{2}\alpha Cos^{2}\alpha =Cos^{2}\alpha(Cos^{2}\alpha +Sin^{2}\alpha)=Cos^{2} \alpha*1=Cos^{2}\alpha\\\\1+tg^{2}\alpha=\frac{1}{Cos^{2}\alpha}\\\\Cos^{2}\alpha=\frac{1}{1+tg^{2}\alpha}=\frac{1}{1+2^{2} }=\frac{1}{5}=0,2$

Answer 2

To simplify the expression cos⁴a + sin²a × cos²a, we can use trigonometric identities to rewrite the terms in a more manageable form.

Trigonometric Identity:

1. sin²a = 1 - cos²a

Now, let's simplify the expression:

cos⁴a + sin²a × cos²a

Substitute sin²a with (1 - cos²a):

cos⁴a + (1 - cos²a) × cos²a

Expand the second term:

cos⁴a + cos²a - cos⁴a

Now, the two cos⁴a terms cancel out:

cos²a

Now that the expression is simplified to cos²a, let's find its value when tan(a) = 2.

Given that tan(a) = 2, we can use the following trigonometric identity:

tan(a) = sin(a) / cos(a)

Since we know that tan(a) = 2, we can set up the following equation:

2 = sin(a) / cos(a)

Now, we can use the Pythagorean identity to relate sin(a) and cos(a):

sin²(a) + cos²(a) = 1

Since we have sin(a) / cos(a) = 2, we can rewrite sin²(a) as (2cos(a))²:

(2cos(a))² + cos²(a) = 1

Expand the equation:

4cos²(a) + cos²(a) = 1

Combine like terms:

5cos²(a) = 1

Now, solve for cos²(a):

cos²(a) = 1 / 5

Finally, find the value of cos(a):

cos(a) = ±√(1 / 5)

Since cosine is positive in the first and fourth quadrants, we take the positive square root:

cos(a) = √(1 / 5)

Now, the simplified expression cos²a is:

cos²a = 1 / 5

Thus, the simplified expression is 1 / 5, and its value when tan(a) = 2 is also 1 / 5.