Question

(c) Prove that : sin A (1 + tan A) + cos A (1 + cot A) = sec A+ cosec A.
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Answer 1

To prove:

sin A (1 + tan A) + cos A (1 + cot A) = sec A + cosec A

Solution:

$\leadsto \: \sf{sinA \bigg(1 + \dfrac{sinA}{cosA}\bigg) + cos A \bigg(1 + \dfrac{cosA}{sinA}\bigg)}$

$= \: \sf{sin A \bigg(\dfrac{cosA+sinA}{cosA}\bigg) + cos A \bigg(\dfrac{sinA+cosA}{sinA}\bigg)}$

$= \: \sf{(sin A + cosA) \bigg[\dfrac{sin A}{cosA}+ \dfrac{cosA}{sinA} \bigg]}$

Taking LCM,

$= \: \sf{(sinA + cosA) \bigg(\dfrac{sin^{2} A + cos^{2} A}{cosA.sinA}\bigg)}$

$= \: \sf{\dfrac{(sinA + cosA)(1)}{cosA.sinA}}$

$= \: \sf{\dfrac{sinA}{cosA.sinA} + \dfrac{cosA}{cosA.sinA}}\\\\\\= \sf{\dfrac{1}{cosA} + \dfrac{1}{sin A}}\\\\\\= \boxed{\bf{secA + cosec A}}\\\\\\\sf{\underline{Hence \: proved}}$