Prove that : = 2(sec A + cosec A)

To prove: = 2(sec A + cosec A) Solving L.H.S. of the above equation : Solving R.H.S. of the equation : Since, L.H.S. = R.H.S. = Hence, proved that = 2(sec A + cosec A).

Mathematics

Prove that :
$\dfrac{\text{1 + cot A}}{\text{cos A}} + \dfrac{\text{1 + tan A}}{\text{sin A}}$ = 2(sec A + cosec A)

Trigonometric Identities

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To prove:

$\dfrac{\text{1 + cot A}}{\text{cos A}} + \dfrac{\text{1 + tan A}}{\text{sin A}}$ = 2(sec A + cosec A)

Solving L.H.S. of the above equation :

$\Rightarrow \dfrac{\text{1 + cot A}}{\text{cos A}} + \dfrac{\text{1 + tan A}}{\text{sin A}} \\[1em] \Rightarrow \dfrac{\text{sin A(1 + cot A) + cos A(1 + tan A)}}{\text{cos A sin A}} \\[1em] \Rightarrow \dfrac{\text{sin A + sin A cot A + cos A + cos A tan A}}{\text{cos A sin A}} \\[1em] \Rightarrow \dfrac{\text{sin A + sin A} \times \dfrac{\text{cos A}}{\text{sin A}} + \text{cos A + cos A} \times \dfrac{\text{sin A}}{\text{cos A}}}{\text{cos A sin A}} \\[1em] \Rightarrow \dfrac{\text{sin A + cos A + cos A + sin A}}{\text{cos A sin A}} \\[1em] \Rightarrow \dfrac{\text{2(sin A + cos A)}}{\text{sin A cos A}}.$

Solving R.H.S. of the equation :

$\Rightarrow \text{2(sec A + cosec A)} \\[1em] \Rightarrow 2\Big(\dfrac{1}{\text{cos A}} + \dfrac{1}{\text{sin A}}\Big) \\[1em] \Rightarrow 2\Big(\dfrac{\text{sin A + cos A}}{\text{sin A cos A}}\Big) \\[1em] \Rightarrow \dfrac{2\text{(sin A + cos A)}}{\text{sin A cos A}}.$

Since, L.H.S. = R.H.S. = $\dfrac{\text{2(sin A + cos A)}}{\text{sin A cos A}}$

Hence, proved that $\dfrac{\text{1 + cot A}}{\text{cos A}} + \dfrac{\text{1 + tan A}}{\text{sin A}}$ = 2(sec A + cosec A).

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Mathematics

Prove that :
$\dfrac{\text{1 + cot A}}{\text{cos A}} + \dfrac{\text{1 + tan A}}{\text{sin A}}$ = 2(sec A + cosec A)

Trigonometric Identities

Answer

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