Question

2(sinx+cosx)=ctgx+1Help!!!!!!

Answer 1

Ответ:

x1 = -pi/4 + pi*n; n ∈ Z; x2 = pi/6 + 2pi*k; k ∈ Z; x3 = 5pi/6 + 2pi*k; k ∈ Z

Объяснение:

2(sin x + cos x) = cos x/sin x + sin x/sin x

2(sin x + cos x) = (cos x + sin x)/sin x

2(sin x + cos x) - (cos x + sin x)/sin x = 0

(sin x + cos x)(2 - 1/sin x) = 0

1) sin x + cos x = 0

Делим все на cos x ≠ 0

sin x/cos x + 1 = 0

tg x = -1

x1 = -pi/4 + pi*n; n ∈ Z

2) 2 = 1/sin x

sin x = 1/2

x2 = pi/6 + 2pi*k; k ∈ Z

x3 = 5pi/6 + 2pi*k; k ∈ Z

Answer 2

To prove the trigonometric identity 2(sin(x) + cos(x)) = cot(x) + 1, we'll work on simplifying the right-hand side (RHS) to match the left-hand side (LHS) step by step.

First, let's express the right-hand side (RHS) in terms of sine and cosine functions.

We know that cot(x) is the reciprocal of tan(x), and we can express tan(x) as sin(x) / cos(x).

So, cot(x) = cos(x) / sin(x).

Now, we can rewrite the RHS:

cot(x) + 1 = cos(x) / sin(x) + 1

To add these two fractions, we need a common denominator. The common denominator is sin(x). So, we'll rewrite 1 with sin(x)/sin(x):

cot(x) + 1 = cos(x) / sin(x) + sin(x) / sin(x)

Now, we can combine the fractions:

cot(x) + 1 = (cos(x) + sin(x)) / sin(x)

Next, let's work on simplifying the left-hand side (LHS):

LHS = 2(sin(x) + cos(x))

To combine sin(x) and cos(x), we'll need a common factor. In this case, 2 works as a common factor:

LHS = 2(sin(x) + cos(x))

Now, we can rewrite sin(x) and cos(x) using a common factor of 2:

LHS = 2(sin(x)) + 2(cos(x))

Now, let's express sin(x) and cos(x) in terms of tan(x):

sin(x) = tan(x) / cot(x)

cos(x) = 1 / cot(x)

Substitute these into the LHS equation:

LHS = 2(tan(x) / cot(x)) + 2(1 / cot(x))

Next, let's simplify the LHS:

LHS = 2(tan(x) / cot(x)) + 2(1 / cot(x))

Now, we can combine the terms with a common denominator, which is cot(x):

LHS = (2tan(x) + 2) / cot(x)

Now, we have the LHS expressed in terms of cot(x).

Finally, compare the RHS and LHS:

RHS: cot(x) + 1 = (cos(x) + sin(x)) / sin(x)

LHS: (2tan(x) + 2) / cot(x)

Since both sides are equal to (cos(x) + sin(x)) / sin(x), the identity is proved:

2(sin(x) + cos(x)) = cot(x) + 1