Prove that Sin A1-tan A-cos A1-cotA- A-coec A

LHS =sinA(1+tan A) +cos A (1+cot A) =sin A ( 1+sin A/cos A )+cos A ( 1+cos A/sin A ) =sin A +sin^2 A/cos A+cos A +sin^2 A/sin A =sin A +cos A +sin^2 A/cos A ...

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Byju's Answer

Standard X

Mathematics

Trigonometric Identities

Prove that Si...

Question

Prove that SinA(1+tanA)+cosA(1+cotA)=secA+coecA

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Solution

LHS =sinA(1+tan A) +cos A (1+cot A)
$= s i n A (1 + \frac{s i n A}{c o s A}) + c o s A (1 + \frac{c o s A}{s i n A}) = s i n A + \frac{s i n^{2} A}{c o s A} + c o s A + \frac{s i n^{2} A}{s i n A} = s i n A + c o s A + \frac{s i n^{2} A}{c o s A} + \frac{c o s^{2} A}{s i n A} = s i n A + c o s A + \frac{s i n^{3} A + c o s^{3} A}{s i n A c o s A}$
$= s i n A + c o s A + \frac{(s i n A + c o s A) (s i c^{2} A - s i n A c o s A + c o s^{2} A)}{s i n A c o s A} (a^{3} + b^{3}) = (a + b) (a^{2} + a b + b^{2}) = (s i n A + c o s A) [1 + \frac{(1 - s i n A c o s a)}{s i n A c o s A}] = (s i n A + c o s A) [\frac{s i n A c o s A + 1 - s i n A c o s A}{s i n A c o s A}] = (s i n A + c o s A) [\frac{1}{s i n A c o s A}] = \frac{s i n A}{s i n A c o s A} + \frac{c o s A}{s i n A c o s A} = \frac{1}{c o s A} + \frac{1}{s i n A} = s i n A + c o s e c A .$

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