Considering the principal values of inverse trigonometric functions, the value of the expression

Q. Considering the principal values of inverse trigonometric functions, the value of the expression

tan ( 2 sin⁻¹ ( 2 / √13 ) − 2 cos⁻¹ ( 3 / √10 ) )

is equal to :

A. 33/56

B. −33/56

C. −16/63

D. 16/63

Correct Answer: 33/56

Explanation

Let

α = sin⁻¹( 2 / √13 )

Since sin⁻¹x lies in the interval [−π/2, π/2], α lies in the first quadrant.

So we take a right-angled triangle with:

sin α = 2 / √13

Hence,

cos α = √(1 − sin²α) = √(1 − 4/13) = 3 / √13

Now,

tan α = (2 / √13) / (3 / √13) = 2/3

Using the identity:

tan 2α = 2 tan α / (1 − tan² α)

we get:

tan 2α = 2(2/3) / (1 − 4/9) = (4/3) / (5/9) = 12/5

Now let

β = cos⁻¹( 3 / √10 )

Since cos⁻¹x lies in [0, π], β lies in the first quadrant.

So,

sin β = √(1 − cos²β) = √(1 − 9/10) = 1 / √10

Hence,

tan β = (1 / √10) / (3 / √10) = 1/3

Using the identity:

tan 2β = 2 tan β / (1 − tan² β)

we get:

tan 2β = 2(1/3) / (1 − 1/9) = (2/3) / (8/9) = 3/4

Now the required expression is:

tan(2α − 2β)

Using the identity:

tan(A − B) = (tan A − tan B) / (1 + tan A tan B)

Substitute tan 2α = 12/5 and tan 2β = 3/4:

tan(2α − 2β) = (12/5 − 3/4) / (1 + (12/5)(3/4))

= (48 − 15)/20 / (1 + 36/20)

= (33/20) / (56/20) = 33/56

Hence, the value of the given expression is

33/56

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