Question

Assertion: The equality tan (cot^–1 x) = cot (tan^–1 x), is true for all x ∈ R.

Reason: The identity tan^–1 x + cot^–1 x = `π/2`, is true for all x ∈ R.

Which of the following is correct?

पर्याय

• Both Assertion and Reason are true and Reason is the correct explanation for Assertion.

• Both Assertion and Reason are true but Reason is not the correct explanation for Assertion.

• Assertion is true and Reason is false.

• Assertion is false and Reason is true.

Answer 1

Both Assertion and Reason are true and Reason is the correct explanation for Assertion.

Explanation:

Assertion: tan (cot^–1 x) = cot (tan^–1 x) for all x ∈ R.

Let cot^–1 x = α

cot α = x

tan α = `1/x`

So, tan (cot^–1 x) = tan α = `1/x`

Also let tan^–1 x = β

tan β = x

cot β = `1/x`

So, cot (tan^–1 x) = cot β = `1/x`

Hence, tan (cot^–1 x) = cot (tan^–1 x) = `1/x` (x ≠ 0)

So the Assertion is true (with the usual understanding that x ≠ 0).

Reason: tan^–1 x + cot^–1 x = `π/2` for all x ∈ R.

This is a standard inverse-trigonometric identity.

From it we get:

cot^–1 x = `π/2 - tan^-1 x`

Then, tan (cot^–1 x) = `tan (π/2 - tan^-1 x)` = cot (tan^–1 x)

which is exactly the assertion.

So the Reason is true and it directly explains why the assertion holds.

Therefore, both Assertion and Reason are true, and Reason is the correct explanation for Assertion.