Question

sinA - sin^3A÷ 2cos^3 - cosA =tanA

Answer 1

♧♧HERE IS YOUR ANSWER♧♧

Trigonometry is the study of angles and its ratios. This study prescribes the relation amongst the sides of a triangle and angles of the triangle.

There are sine, cosine, tangent, cot, sectant and cosec ratios to an angle of a triangle.

Let me tell you an interesting fact about Trigonometry.

"Triangle" > "Trigonometry"

Remember some formulae now :

sin²θ + cos²θ = 1

sec²θ - tan²θ = 1

cosec²θ - cot²θ = 1

Want to learn more!

Here it is :

sin(A + B) = sinA cosB + cosA sinB

sin(A - B) = sinA cosB - cosA sinB

cos(A + B) = cosA cosB - sinA sinB

cos(A - B) = cosA cosB + sinA sinB

SOLUTION :

L.H.S.

$\frac{sin \alpha - 2 {sin}^{3} \alpha }{2 {cos}^{3} \alpha - cos \alpha } \\ \\ = \frac{sin \alpha (1 - 2 {sin}^{2} \alpha )}{cos \alpha (2 {cos}^{2} \alpha - 1) } \\ \\ = \frac{sin \alpha \times cos2 \alpha }{cos \alpha \times cos2 \alpha } \\ \\ = \frac{sin \alpha }{cos \alpha } \\ \\ = tan \alpha$

= R.H.S. [Proved]

♧♧HOPE IT HELPS YOU♧♧

Answer 2

Trigonometry is the study of the relationship between the sides and angles of a triangle.
An equation involving trigonometry ratios of an angle is called is called a trigonometric identity, if it is true for all values of the angles involved. For any acute angle θ, we have the following identities.
i) sin² θ + cos² θ = 1 , ii) 1 + tan² θ = sec² θ , iii) cot² θ +1 = cosec² θ, iv) tan θ = sin θ/cos θ , v) cot θ = cos θ / sin θ.
SOLUTION:
Mistake in the question: It is 2sin³A
Given:
SinA - 2sin³A/ 2cos³ - cosA =tanA
LHS:
= sin A ( 1- 2sin²A)/cosA(2cos²A -1)
= sinA[1-2(1-cos²A)] / cosA(2cos²A -1)
[ sin²A + cos²A= 1 sin²A= 1-cos²A]
= sinA[1-2 + 2cos²A)] / cosA(2cos²A -1)
= sinA[2cos²A - 1] / cosA(2cos²A -1)
= sin A / cos A = tan A = RHS
[ tan A = sinA/cosA]
LHS = RHS

HOPE THIS WILL HELP YOU....