A = (sin θ cos θ tan θ + sin θ cos θ cot θ)

the value of A is ______.

Find the Value of A = sin θ cos θ tan θ + sin θ cos θ cot θ

Understanding the Given Trigonometric Expression

The given question asks us to simplify a trigonometric expression containing sine, cosine, tangent, and cotangent functions. At first glance, the expression may appear lengthy because four different trigonometric functions are involved. However, the expression becomes very simple when tan θ and cot θ are written in terms of sin θ and cos θ.

The given expression is:

A = sin θ cos θ tan θ + sin θ cos θ cot θ

The most direct approach is to simplify each term separately. For this purpose, we use the standard identities for tangent and cotangent.

Trigonometric Identities Required to Solve the Problem

The tangent of an angle is defined as the ratio of sine to cosine:

tan θ = sin θ / cos θ

Similarly, the cotangent of an angle is defined as the ratio of cosine to sine:

cot θ = cos θ / sin θ

We will substitute these two fundamental trigonometric identities into the given expression. This substitution allows the common sine and cosine factors to cancel, reducing the expression to a familiar Pythagorean identity.

Step-by-Step Solution

Step 1: Write the Given Expression

We begin with:

A = sin θ cos θ tan θ + sin θ cos θ cot θ

Instead of trying to simplify the entire expression at once, we can consider the two terms separately. The first term contains tan θ, while the second term contains cot θ.

Step 2: Substitute the Identity for tan θ

Using:

tan θ = sin θ / cos θ

the first term becomes:

sin θ cos θ tan θ = sin θ cos θ × (sin θ / cos θ)

The factor cos θ in the numerator cancels with cos θ in the denominator. Therefore:

sin θ cos θ tan θ = sin θ × sin θ

Hence:

sin θ cos θ tan θ = sin² θ

Step 3: Substitute the Identity for cot θ

Now consider the second term:

sin θ cos θ cot θ

Using:

cot θ = cos θ / sin θ

we obtain:

sin θ cos θ cot θ = sin θ cos θ × (cos θ / sin θ)

The factor sin θ in the numerator cancels with sin θ in the denominator. Therefore:

sin θ cos θ cot θ = cos θ × cos θ

Hence:

sin θ cos θ cot θ = cos² θ

Step 4: Combine the Simplified Terms

We have shown that the first term simplifies to sin² θ and the second term simplifies to cos² θ. Therefore, the original expression becomes:

A = sin² θ + cos² θ

Now we use one of the most fundamental identities in trigonometry:

sin² θ + cos² θ = 1

Therefore:

A = 1

Complete Simplification in a Single Calculation

The entire calculation can also be written compactly as:

A = sin θ cos θ tan θ + sin θ cos θ cot θ

= sin θ cos θ × (sin θ / cos θ) + sin θ cos θ × (cos θ / sin θ)

= sin² θ + cos² θ

= 1

Thus, the complicated-looking expression reduces directly to 1 after applying the basic definitions of tangent and cotangent.

Why the Expression Becomes sin² θ + cos² θ

The structure of the expression is designed so that the denominator present in tan θ or cot θ cancels with one of the factors already multiplying it. In the first term, tan θ contributes sin θ / cos θ, so the existing cos θ factor is cancelled, leaving sin² θ.

In the second term, cot θ contributes cos θ / sin θ, so the existing sin θ factor is cancelled, leaving cos² θ. The two remaining terms then form the standard Pythagorean identity sin² θ + cos² θ = 1.

Understanding the Pythagorean Trigonometric Identity

The identity sin² θ + cos² θ = 1 is one of the most important relationships in trigonometry. It follows from the equation of the unit circle, x² + y² = 1, where the coordinates of a point corresponding to an angle θ are cos θ and sin θ.

Substituting x = cos θ and y = sin θ into the unit-circle equation gives:

cos² θ + sin² θ = 1

This identity holds for every angle for which the original expression is defined. Therefore, after simplification, the value of A is independent of the particular value of θ.

Verification with a Specific Angle

We can verify the result by choosing an angle for which both tan θ and cot θ are defined. Let θ = 45°. Then:

sin 45° = 1/√2,   cos 45° = 1/√2,   tan 45° = 1,   cot 45° = 1

Substituting these values into the original expression:

A = (1/√2)(1/√2)(1) + (1/√2)(1/√2)(1)

A = 1/2 + 1/2

Therefore:

A = 1

This numerical verification agrees with the result obtained through trigonometric simplification.

Final Answer

The value of A = sin θ cos θ tan θ + sin θ cos θ cot θ is 1.