Question

If cos A/ tana.sina= 16 then prove that sec = +- √17 / 4​

Answer 1

answer attached in photo.

Answer 2

The given equation cos A / (tan A · sin A) = 16 is simplified using trigonometric identities, leading to proving that sec A = ± √17 / 4.

Step-by-step explanation:

If the equation cos A / (tan A · sin A) equals 16, we are asked to prove that sec A equals ± √17 / 4.

First, let's simplify the given equation using trigonometric identities:

• tan A = sin A / cos A
• sec A = 1 / cos A

Substituting the identity of tan A into the equation, we get:

cos A / ((sin A / cos A) · sin A) = cos^2 A / sin^2 A

Therefore, the equation becomes:

cos^2 A / sin^2 A = 16

Which can be further written as:

(1 / sin^2 A) - 1 = 16

Simplifying, we have:

csc^2 A - 1 = 16

Knowing that csc A = 1 / sin A and using the Pythagorean identity csc^2 A - 1 = cot^2 A, we rewrite the equation:

cot^2 A = 16

Since cot A = cos A / sin A, the equation further simplifies to:

cos^2 A / sin^2 A = 16

Using another identity sin^2 A + cos^2 A = 1, we get:

cos^2 A = 16 sin^2 A

cos^2 A = 16 (1 - cos^2 A)

Which leads us to:

17 cos^2 A = 16

cos^2 A = 16 / 17

Eventually, sec A = 1 / cos A gives us:

sec A = 1 / (± √(cos^2 A))

sec A = ± √17 / 4

This proves the required result.