Question

Cota -tana=2cos^2a-1/sina cosa

Answer 1

Question :

Prove that,

$\sf{cot\:a-tan\:a=\dfrac{2cos^2a-1}{sin\:a\:.\:cos\:a}}$

Solution :

Taking L.H.S,

$\sf{cot\:a\:-\:tan\:a}$

$\implies\sf{\dfrac{cos\:a}{sin\:a}-\dfrac{sin\:a}{cos\:a}}$

$\implies\sf{\dfrac{cos^2a-sin^2a}{sin\:a\:.\:cos\:a}}$

Use the Identity : cos²A - sin²A = cos2A

$\implies\sf{\dfrac{cos2a}{sin\:a\:.\:cos\:a}}$

Use the identity : cos2A =2cos²A- 1

$\implies\sf{\dfrac{2cos^2A-1}{sin\:a\:.\:cos\:a}}$

L.H.S = R.H.S [ Proved ]

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More identities :-

• sin2A = 2sinA cosA

• 1 + cos2A = 2cos²A

• cos2A = 1 - 2sin²A

• 1 - cos2A = 2sin²A

• $\sf{sin2A=\dfrac{2tanA}{1+tan^2A}}$

•$\sf{tan2A=\dfrac{2tanA}{1-tan^2A}}$

• sin3A = 3sinA - 4sin³A

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Answer 2

$\huge \mathfrak{Prove \: that:-}$

$\implies$ cot A - tan A = 2cos²A - 1 / sin A . cos A

$\large\underline\mathrm{Solution}$

$\large\underline\mathrm{LHS}$

$\implies$ cot A - tan A

$\implies$ cos A/sin A - sin A/ cos A

$\implies$ cos²A - sin²A / sin A cos A

$\implies$ cos2A/sin A.cos A [cos²A - sin²A = cos2A]

$\implies$ 2cos²A - 1 / sin A . cos A [cos2A = 2cos²A - 1]

$\huge \mathfrak{Basic \: Information:-}$

$\implies$ sin2A . 2sin A cos A

$\implies$ 1 + cos2A = 2cos2A

$\implies$ sin2A . 2tan A/1 + tan²a

$\implies$ sin3A = 3sinA - 4sin³A

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