Solveeit Logo
Login

Question

Question: Find the value of trigonometric equation \({\text{tan38}}^\circ - {\text{cot22}}^\circ \) = ? \( ...

Find the value of trigonometric equation tan38∘−cot22∘{\text{tan38}}^\circ - {\text{cot22}}^\circ = ?
A. 12cosec 38∘ sec 22∘ B. 2 sin 22∘ cos 38∘ C. - 12cosec 22∘ sec 38∘ D. none of these  {\text{A}}{\text{. }}\dfrac{1}{2}{\text{cosec 38}}^\circ {\text{ sec 22}}^\circ \\\ {\text{B}}{\text{. 2 sin 22}}^\circ {\text{ cos 38}}^\circ \\\ {\text{C}}{\text{. - }}\dfrac{1}{2}{\text{cosec 22}}^\circ {\text{ sec 38}}^\circ \\\ {\text{D}}{\text{. none of these}} \\\

Explanation

Solution

Hint: To solve the trigonometric equation given in the question we convert the tan and cot functions in terms of sin and cos functions and simplify using basic trigonometric identities for answer.
Complete step-by-step answer:
tan38∘−cot22∘{\text{tan38}}^\circ - {\text{cot22}}^\circ = sin 38∘cos⁡38∘\dfrac{{{\text{sin 38}}^\circ }}{{\cos 38^\circ }} - cos 22∘sin⁡22∘\dfrac{{{\text{cos 22}}^\circ }}{{\sin 22^\circ }} -- (tanθ = sin⁡θcos⁡θ\dfrac{{\sin \theta }}{{\cos \theta }} and cotθ =cos⁡θsin⁡θ\dfrac{{\cos \theta }}{{\sin \theta }})
⟹tan38∘−cot22∘{\text{tan38}}^\circ - {\text{cot22}}^\circ = sin 38∘sin 22∘−cos⁡ 22∘cos⁡ 38∘cos⁡38∘sin 22∘\dfrac{{{\text{sin 38}}^\circ {\text{sin 22}}^\circ - \cos {\text{ 22}}^\circ \cos {\text{ 38}}^\circ }}{{\cos 38^\circ {\text{sin 22}}^\circ }} --- (1)
We know that,
cos (A + B) = cos A cos B – sin A sin B
Using the above equation in equation (1), we get
⟹tan38∘−cot22∘{\text{tan38}}^\circ - {\text{cot22}}^\circ =  - cos(38+22)∘cos⁡38∘sin 22∘\dfrac{{{\text{ - cos}}\left( {38 + 22} \right)^\circ }}{{\cos 38^\circ {\text{sin 22}}^\circ }}
⟹tan38∘−cot22∘{\text{tan38}}^\circ - {\text{cot22}}^\circ =  - cos(60)∘cos⁡38∘sin 22∘\dfrac{{{\text{ - cos}}\left( {60} \right)^\circ }}{{\cos 38^\circ {\text{sin 22}}^\circ }}
⟹tan38∘−cot22∘{\text{tan38}}^\circ - {\text{cot22}}^\circ =  - 12cos⁡38∘sin 22∘\dfrac{{{\text{ - }}\dfrac{1}{2}}}{{\cos 38^\circ {\text{sin 22}}^\circ }}
⟹tan38∘−cot22∘{\text{tan38}}^\circ - {\text{cot22}}^\circ = −12cosec 22∘ sec 38∘- \dfrac{1}{2}{\text{cosec 22}}^\circ {\text{ sec 38}}^\circ
Hence, Option C is the correct answer.

Note: In order to solve this type of questions the key is to convert the given trigonometric functions into a different form such that it can be simplified using a known trigonometric identity. Then we reduce it into another simple trigonometric value or a function using appropriate formulae, to determine the answer. Basic knowledge of trigonometric functions and identities is essential.