Simplifying Trigonometric Expressions: Solving sin[2 tan^-1 1/3]cos[tan^-1 2√2]

Trigonometric expressions often require the application of various identities to simplify and evaluate them. This article will walk through the process of simplifying the expression sin[2 tan^-1 1/3]cos[tan^-1 2√2].

Using Trigonometric Identities to Simplify

The given expression is:

sin[2 tan^-1 1/3]cos[tan^-1 2√2]

Step 1: Simplifying (cos[tan^-1 2√2])

To simplify cos[tan^-1 2√2], we use the identity:

cos[tan^-1 x] 1/√(1 x^2)

Since (x 2√2), we have:

cos[tan^-1 2√2] 1/√(1 (2√2)^2) 1/√(1 8) 1/√9 1/3

Step 2: Simplifying (sin[2 tan^-1 1/3])

Using the double angle identity for sine:

sin[2θ] 2sinθcosθ

Let tan^-1(1/3) θ. Then:

sinθ 1/√10 and cosθ 3/√10

Substituting these values into the double angle identity:

sin[2θ] 2(1/√10)(3/√10) 6/10 3/5

Putting It All Together

Now, the expression sin[2 tan^-1 1/3]cos[tan^-1 2√2] simplifies to:

(3/5)(1/3) 3/15 1/5

Therefore, the value of the given expression is:

1/5

Another Approach Using Trigonometric Identities

We can also use the tangent and cosine inverse identities to simplify the given expression:

Let - sin [2 tan^-11/3] cos[tan^-12√2] x ………. 1

If tan^-1 1/3 θ, we have:

tan θ 1/3

To find sine and cosine of θ, we can use the Pythagorean theorem:

Let the opposite side be 1 and adjacent side be 3, then the hypotenuse h is:

h^2 1^2 3^2 1 9 10

Therefore, sin θ 1/√10 and cos θ 3/√10

Using the double angle formula for sine:

sin 2θ 2sin θ cos θ 2(1/√10)(3/√10) 6/10 3/5

So, sin [2 tan^-11/3] 3/5

Next, let tan^-12√2 φ. Using the identity:

cos [tan^-1 x] 1/√(1 x^2)

We find:

cos [tan^-12√2] cos φ 1/√(1 (2√2)^2) 1/√(1 8) 1/3

Substituting these values into the original expression:

sin [2 tan^-11/3] cos [tan^-12√2] 3/5 * 1/3 3/15 1/5

Therefore, the simplified expression is:

1/5