Prove that-1- A -tan A -sin A -cos A - 3 A -cosec3 A - 2 A -cosec2 A

L.H.S = (1+cot A+tan A).(sin A cos A) = ( 1+cos A/sin A+sin A/cos A )(sin A cos A) = ( sin A cos A+cos^2A+sin^2A/sin A.cos A )(sin A cos A) =sin^3A cos^3A/si ...

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Byju's Answer

Standard X

Mathematics

Relation between Trigonometric Ratios

Prove that:1+...

Question

Prove that:
$(1 + c o t A + t a n A) . (s i n A - c o s A) = \frac{s e c^{3} A - c o s e c^{3} A}{s e c^{2} A . c o s e c^{2} A}$

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Solution

L.H.S = $(1 + c o t A + t a n A) . (s i n A - c o s A)$

$= (1 + \frac{c o s A}{s i n A} + \frac{s i n A}{c o s A}) (s i n A - c o s A)$

$= (\frac{s i n A c o s A + c o s^{2} A + s i n^{2} A}{s i n A . c o s A}) (s i n A - c o s A)$

$= \frac{s i n^{3} A - c o s^{3} A}{s i n A . c o s A}$

[Using $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ ]

$= \frac{\frac{s i n^{3} A}{s i n^{3} A . c o s^{3} A} - \frac{c o s^{3} A}{s i n^{3} A . c o s^{3} A}}{\frac{s i n A c o s A}{s i n^{3} A . c o s^{3} A}}$

[Dividing numerator and denominator by $s i n^{3} . A c o s^{3} A$ ]

$= \frac{s e c^{3} A - c o s e c^{3} A}{s e c^{2} A . c o s e c^{2} A}$ =R.H.S

Suggest Corrections

Prove that: (1+cot A+tan A).(sin A−cos A)=sec3A−cosec3Asec2A.cosec2A

Prove that:
$(1 + c o t A + t a n A) . (s i n A - c o s A) = \frac{s e c^{3} A - c o s e c^{3} A}{s e c^{2} A . c o s e c^{2} A}$