KnowledgeBoat Logo
|

Mathematics

Prove the following identities, where the angles involved are acute angles for which the trigonometric ratios are defined:

cot A - tan A=2 cos2A−1sin A cos A\text{cot A - tan A} = \dfrac{2\space\text{cos}^2 A - 1}{\text{sin A cos A}}

Trigonometric Identities

71 Likes

Answer

The L.H.S of the equation can be written as,

⇒cos Asin A−sin Acos A⇒cos2A−sin2Asin A cos A⇒cos2A−(1 - cos2A)sin A cos A⇒cos2A+cos2A−1sin A cos A⇒2cos2A−1sin A cos A\Rightarrow \dfrac{\text{cos A}}{\text{sin A}} - \dfrac{\text{sin A}}{\text{cos A}} \\[1em] \Rightarrow \dfrac{\text{cos}^2 A - \text{sin}^2 A}{\text{sin A cos A}} \\[1em] \Rightarrow \dfrac{\text{cos}^2 A - \text{(1 - cos}^2 A)}{\text{sin A cos A}} \\[1em] \Rightarrow \dfrac{\text{cos}^2 A + \text{cos}^2 A - 1}{\text{sin A cos A}} \\[1em] \Rightarrow \dfrac{2\text{cos}^2 A - 1}{\text{sin A cos A}} \\[1em]

Since, L.H.S. = R.H.S. hence, proved that cot A - tan A = 2cos2A−1sin A cos A\dfrac{2\text{cos}^2 A - 1}{\text{sin A cos A}}.

Answered By

24 Likes

Related Questions