Question

1)cos53*cos8+sin53*sin8 2)cos^2(2a)+4sin^2(a)*cos^2(a)3)cos2a(1+tg^2(a))-1С решением пж

Answer 1

1)=cos(53-8)=cos45°=√2/2.

2)=cos²2α+(2sinαcosα)²=cos²2α+sin²2α=1.

3)=cos2α·1/cos²α-1= (cos2α-1) /cos²α= - (1-cos2α) /cos²α=-2sin²α/cos²α=-2tg²α.

Answer 2

Sure, I can help you with those trigonometric expressions. Let's solve each of them step-by-step:

1. cos(53) * cos(8) + sin(53) * sin(8)

We can use the trigonometric identity: cos(a - b) = cos(a) * cos(b) + sin(a) * sin(b)

So, the expression becomes: cos(53 - 8)

Now, we have: cos(45)

And since cos(45) = 1 / √2 or approximately 0.7071, the answer is approximately 0.7071.

2. cos^2(2a) + 4sin^2(a) * cos^2(a)

We can use the Pythagorean identity: sin^2(x) + cos^2(x) = 1

So, the expression becomes: cos^2(2a) + 4(1 - cos^2(a)) * cos^2(a)

Expand the expression: cos^2(2a) + 4cos^2(a) - 4cos^4(a)

Now, use the double angle identity: cos(2a) = 2cos^2(a) - 1 Replace cos^2(2a) with the double angle identity: (2cos^2(a) - 1) + 4cos^2(a) - 4cos^4(a)

Combine like terms: 6cos^2(a) - 4cos^4(a) - 1

3. cos(2a)(1 + tan^2(a)) - 1

We can use the trigonometric identity: tan^2(x) + 1 = sec^2(x)

So, the expression becomes: cos(2a) * sec^2(a) - 1

Now, we know that: sec^2(x) = 1/cos^2(x), so we can replace sec^2(a) with 1/cos^2(a): cos(2a) * (1/cos^2(a)) - 1

Simplify: cos(2a) / cos^2(a) - 1

Now, use the double angle identity for cosine: cos(2a) = 2cos^2(a) - 1 Replace cos(2a) with the double angle identity: (2cos^2(a) - 1) / cos^2(a) - 1

Now, divide each term by cos^2(a): 2 - 1 / cos^2(a) - 1

Finally, simplify: 1 / cos^2(a)

Since 1 / cos^2(a) is the reciprocal of cos^2(a), it is equal to sec^2(a). Therefore, the expression simplifies to sec^2(a).

So, the final answers are:

1. Approximately 0.7071
2. 6cos^2(a) - 4cos^4(a) - 1
3. sec^2(a)