Question

Prove that, $\dfrac{{1 + \tan A}}{{\sin A}} + \dfrac{{1 + \cot A}}{{\cos A}} = 2(\sec A + \cos ecA)$ ?

Answer 1

Here we are given a trigonometric equation and we need to verify whether the given equation is true or not. To prove the equation, we need to solve the left-hand side equation. To solve this, we will use some of the trigonometric formulas i.e. the relation between trigonometric ratios. And then we will try to simplify the expression. At the end we will get the required answer which is our RHS.

Formula used:
a) The following reciprocal identities are to be used.
$\dfrac{1}{{\sin \theta }} = \cos ec\theta$
$\dfrac{1}{{\cos \theta }} = \sec \theta$
b) The following quotient identities are to be used.
$\tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }}$
$\cot \theta = \dfrac{{\cos \theta }}{{\sin \theta }}$

Complete step-by-step answer:
The given trigonometric equation is $1 + \dfrac{{\tan A}}{{\sin A}} + 1 + \dfrac{{\cot A}}{{\cos A}} = 2\cos ecA$
First, let us consider and solve the left-hand side of the equation.
$\dfrac{{1 + \tan A}}{{\sin A}} + \dfrac{{1 + \cot A}}{{\cos A}}$
Now we shall split all the terms present on the numerators.
$\dfrac{{1 + \tan A}}{{\sin A}} + \dfrac{{1 + \cot A}}{{\cos A}} = \dfrac{1}{{\sin A}} + \dfrac{{\tan A}}{{\sin A}} + \dfrac{1}{{\cos A}} + \dfrac{{\cot A}}{{\cos A}}$
$= \cos ecA + \dfrac{{\tan A}}{{\sin A}} + \sec A + \dfrac{{\cot A}}{{\cos A}}$ (Here we have applied $\dfrac{1}{{\sin \theta }} = \cos ec\theta$ and $\dfrac{1}{{\cos \theta }} = \sec \theta$ ) = \cos ecA + \tan A\dfrac{1}{{\sin A}} + \sec A + \cot A\dfrac{1}{{\cos A}}$$$ = \cos ecA + \dfrac{{\sin A}}{{\cos A}}\dfrac{1}{{\sin A}} + \sec A + \dfrac{{\cos A}}{{\sin A}}\dfrac{1}{{\cos A}}$$ (Now we have applied the quotient identities) $$ = \cos ecA + \dfrac{1}{{\cos A}} + \sec A + \dfrac{1}{{\sin A}}$$ (Here we have applied\dfrac{1}{{\sin \theta }} = \cos ec\theta $and$\dfrac{1}{{\cos \theta }} = \sec \theta $ )
$= \cos ecA + \sec A + \sec A + \cos ecA$
$= 2\cos ecA + 2\sec A$
$= 2\left( {\cos ecA + \sec A} \right)$
$= 2\left( {secA + co\sec A} \right)$
Since the answer matches the right-hand side, the equation is a true statement.

Note: Generally, trigonometric identities are equalities that involve trigonometric functions and are useful whenever trigonometric functions are involved in an expression or an equation and the basic trigonometric ratios are sine, cosine, tangent, cosecant, secant, and cotangent, and all the fundamental trigonometric identities are derived from the six trigonometric ratios.
We need to apply the appropriate trigonometric identities to obtain the required answer.