Question

Lg^2(tg^2x)+lg(cos x)=lg(sinx)

Answer 1

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Answer 2

To solve the equation lg^2(tg^2x) + lg(cos x) = lg(sinx), we need to find the value(s) of x that satisfy the equation. Here, lg represents the logarithm base 10, and tg is the tangent function.

Let's break down the steps to solve the equation:

Step 1: Simplify the equation Using logarithm properties, lg^2(tg^2x) can be rewritten as (lg(tg^2x))^2, and lg(sinx) can be rewritten as (lg(sin x))^2. Thus, the equation becomes:

(lg(tg^2x))^2 + lg(cos x) = (lg(sin x))^2

Step 2: Use change of base formula To make it easier to work with trigonometric functions and logarithms, let's convert the equation into a common base. The natural logarithm (ln) will be useful here. The change of base formula is:

lg(a) = ln(a) / ln(10)

Therefore, the equation can be rewritten as:

(ln(tg^2x) / ln(10))^2 + ln(cos x) / ln(10) = (ln(sin x) / ln(10))^2

Step 3: Eliminate denominators To simplify the equation further, let's remove the denominators by multiplying both sides by (ln(10))^2:

(ln(tg^2x))^2 + ln(cos x) * (ln(10))^2 = (ln(sin x))^2 * (ln(10))^2

Step 4: Simplify the equation Let a = ln(tg^2x) and b = ln(cos x), and c = ln(sin x), then the equation becomes:

a^2 + (ln(10))^2 * b = (ln(10))^2 * c

Step 5: Solve for a From the equation a^2 + (ln(10))^2 * b = (ln(10))^2 * c, we can isolate a:

a^2 = (ln(10))^2 * c - (ln(10))^2 * b

a^2 = (ln(10))^2 * (c - b)

a = ±√((ln(10))^2 * (c - b))

Step 6: Solve for x Since a = ln(tg^2x), we have:

ln(tg^2x) = ±√((ln(10))^2 * (c - b))

Now, rewrite it back using the properties of logarithms:

tg^2x = e^(±√((ln(10))^2 * (c - b)))

tg^2x = e^(±√(ln(10))^2 * (c - b))

tg^2x = e^(±ln(10) * √(c - b))

tg^2x = e^(ln(10) * ±√(c - b))

Now, remove the square of the tangent:

tgx = ±e^(ln(10) * ±√(c - b))

Step 7: Final step Now, solve for x by taking the inverse tangent (tan^-1) of both sides:

x = ±tan^-1(e^(ln(10) * ±√(c - b)))

Keep in mind that there might be multiple solutions for x depending on the values of c and b. Without specific numerical values for c and b, the solution will remain in this general form.