Question

prove that sin 7tita - sin 5tita/ cos7 tita + cos5 tita =tan tita​

Answer 1

EXPLANATION.

prove that = sin7∅ - sin5∅/cos7∅ + cos5∅ = tan∅

Using formula,

(a) = sin C - sin D = 2cos ( c + d )/2 sin ( c - d )/2.

(b) = cos C + cos D = 2cos ( c + d )/2 cos ( c - d )/2.

put the formula in equation. we get,

$\sf\implies \dfrac{2 cos \dfrac{( 7\theta + 5\theta )}{2}sin \dfrac{( 7\theta - 5\theta )}{2} }{2 cos \dfrac{( 7\theta + 5\theta )}{2}cos \dfrac{( 7\theta - 5\theta )}{2} }$

$\sf \implies\dfrac{2cos\dfrac{(12\theta)}{2}sin \dfrac{(2\theta )}{2} }{2cos\dfrac{( 12\theta)}{2}cos\dfrac{(2\theta)}{2} }$

$\sf\implies \dfrac{2cos (6\theta) sin(\theta)}{2cos (6\theta ) cos (\theta )}$

⇒ sin∅/cos∅ = tan∅

HENCE PROVED.

                                           

MORE INFORMATION.

TRIGONOMETRIC EQUATION.

(a) = sin∅ = 0 ⇒ ∅ = nπ where n ∈ I

(b) = cos∅ = 0 ⇒ ∅ = nπ + π/2 where n ∈ I.

(c) = sin∅ = 1 ⇒ ∅ = 2nπ + π/2 where n ∈ I.

(d) = sin∅ = -1 ⇒ ∅ = 2nπ - π/2 where n ∈ I.

(e) = cos∅ = 1 ⇒ ∅ = 2nπ where n ∈ I.

(f) = cos∅ = -1 ⇒ ∅ = ( 2n + 1)π where n ∈ I.

(g) = sin∅ = ± 1 ⇒ ∅ = ( 2n + 1 )π/2 where n ∈ I.

(h) = cos∅ = ± 1 ⇒ ∅ = nπ where n ∈ I.

Answer 2

Answer:

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