Question

if theta is an acute angle such that tan theta =2/3,then evaluate 1+tantheta/sin theta+costheta*1-cot theta/sec thta +coec theta

Answer 1

Given:

theta is an acute angle such that tan theta =2/3,

To find:

evaluate 1+tantheta/sin theta+costheta*1-cot theta/sec theta +coec theta

Solution:

From given, we have,

tan ∅ = 2/3

We have the expression,

(1 + tan ∅)/(sin ∅ + cos ∅) × (1 - cot ∅)/(sec ∅ + cosec ∅)

we will use the Pythagorean triplets to solve this problem.

tan ∅ = 2/3

⇒ sin ∅ = 2/5

⇒ cosec ∅ = 5/2

⇒ cos ∅ = 3/5

⇒ sec ∅ = 5/3

⇒ cot ∅ = 3/2

now substitute these values in the above equation.

So, we get,

(1 + tan ∅)/(sin ∅ + cos ∅) × (1 - cot ∅)/(sec ∅ + cosec ∅)

= (1 + 2/3)/(2/5 + 3/5) × (1 - 3/2)/(5/3 + 5/2)

upon solving the above equation, we get,

(1 + tan ∅)/(sin ∅ + cos ∅) × (1 - cot ∅)/(sec ∅ + cosec ∅) = - 1/5