fill in the blanks:
(sin x + sin 3x)/ (cos x + cos 3x) =_____
Answers
EXPLANATION.
⇒ (sin x + sin 3x)/(cos x + cos 3x).
As we know that,
Formula of :
⇒ sin3x = 3sinx - 4sin³x.
⇒ cos3x = 4cos³x - 3cosx.
Using this formula in this equation, we get.
⇒ (sin x + 3sinx - 4sin³x)/(cos x + 4cos³x - 3cosx).
⇒ (4sinx - 4sin³x)/(4cos³x - 2cosx).
⇒ 4sinx(1 - sin²x)/2cosx(2cos²x - 1).
As we know that,
Formula of :
⇒ sin²x + cos²x = 1.
⇒ cos²x = 1 - sin²x.
⇒ cos2x = 2cos²x - 1.
Using this formula in this equation, we get.
⇒ 4sinx(cos²x)/2cosx(cos2x).
⇒ 2sinx(cos²x)/cos x(cos2x).
⇒ 2sinx x cos(x) x cos(x)/cos x(cos2x).
⇒ 2sinx.cosx/cos2x.
As we know that,
Formula of :
⇒ sin2x = 2sinx.cosx.
Using this formula in the equation, we get.
⇒ sin2x/cos2x = tan2x.
MORE INFORMATION.
Trigonometric ratios of multiple angles.
(1) = sin2θ = 2sinθ.cosθ.
(2) = cos2θ = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ = 1 - tan²θ/1 + tan²θ.
(3) = tan2θ = 2tanθ/1 - tan²θ.
(4) = sin3θ = 3sinθ - 4sin³θ.
(5) = cos3θ = 4cos³θ - 3cosθ.
(6) = tan3θ = 3tanθ - tan³θ/1 - 3tan²θ.