Question

If tanβ=cosθtanα , then cot²(θ/2)=​

Answer 1

EXPLANATION.

⇒ tanβ = cosθ.tanα.

As we know that,

We can write equation as,

⇒ cosθ = tanβ/tanα.

As we know that,

Formula of :

⇒ cos2θ = 1 - tan²θ/1 + tan²θ.

We can write = cosθ as,

⇒ cosθ = 1 - tan²(θ/2)/1 + tan²(θ/2).

⇒ [1 + tan²(θ/2)](cosθ) = 1 - tan²(θ/2).

⇒ cosθ + cosθ.tan²(θ/2) = 1 - tan²(θ/2).

⇒ cosθ.tan²(θ/2) + tan²(θ/2) = (1 - cosθ)

⇒ tan²(θ/2)[1 + cosθ] = [1 - cosθ].

As we know that,

⇒ Value of cosθ = tanβ/tanα.

⇒ Put the value in the equation, we get.

⇒ tan²(θ/2)[1 + tanβ/tanα] = [1 - tanβ/tanα].

⇒ tan²(θ/2)[tanα + tanβ/tanα] = [tanα - tanβ/tanα].

⇒ tan²(θ/2)[tanα + tanβ] = [tanα - tanβ].

As we know that,

Formula of :

⇒ tanθ = sinθ/cosθ.

Using this formula in equation, we get.

⇒ tan²(θ/2)[sinα/cosα + sinβ/cosβ] = [sinα/cosα - sinβ/coβ].

⇒ tan²(θ/2)[sinα.cosβ + cosα.sinβ/cosα.cosβ] = [sinα.cosβ - cosα.sinβ/cosα.cosβ].

⇒ tan²(θ/2) [sinα.cosβ + cosα.sinβ] = [sinα.cosβ - cosα.sinβ].

As we know that,

Formula of :

⇒ sin (α ± β) = sinα.cosβ ± cosα.sinβ.

Using this formula in equation, we get.

⇒ tan²(θ/2)[sin(α + β)] = [sin(α - β)].

⇒ tan²(θ/2) = [sin(α - β)]/[sin(α + β)].

As we know that,

⇒ tanθ = 1/cotθ.

Using this formula in equation, we get.

⇒ 1/cot²(θ/2) =  [sin(α - β)]/[sin(α + β)].

⇒ [sin(α - β)][cot²(θ/2)] = [sin(α + β)].

⇒ cot²(θ/2) = [sin(α + β)]/[sin(α - β)].

                                                                                                                                 

MORE INFORMATION.

Properties of inverse trigonometric functions.

(1) = sin⁻¹(sinθ) = θ  provided that = -π/2 ≤ θ ≤ π/2.

(2) = cos⁻¹(cosθ) = θ provided that = 0 ≤ θ ≤ π.

(3) = tan⁻¹(tanθ) = θ provided that = -π/2 < θ < π/2.

(4) = cot⁻¹(cotθ) = θ provided that = 0 < θ < π.

(5) = sec⁻¹(secθ) = θ provided that = 0 ≤ θ < π/2  Or  π/2 < θ ≤ π.

(6) = cosec⁻¹(cosecθ) = θ provided that = -π/2 ≤ θ < 0  Or  0 < θ ≤ π/2.

Answer 2

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Formula for this question :-

cos2θ = 1-tan ² θ / 1 + tan ² θ

• cosθ

cos θ = 1-tan²(θ/2) / 1 + tan² (θ/2)

• [1 + tan² (θ/2) ] (cosθ) = 1- tan² (θ/2)
• cosθ + cosθ.tan²(θ/2) = 1 - tan²(θ/2)
• cosθ.tan²(θ/2) + tan²(θ/2) = (1 - cosθ)
• tan²(θ/2) [1 + cosθ] = [1 - cos θ]

Value of cos θ = tanβ / tanα

• tan²(θ/2)[1 + tanβ/tanα] = [1-tanβ/tanα].
• tan²(θ/2)[tanα + tanβ/tanα] = [tanα - tanβ/tanα]
• tan² (θ/2) [tanα + tanβ] = [tanα - tanβ]

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Formula tan θ = sinθ / cosθ

• tan²(θ/2)[sinα/cosα + sinβ/cosβ] = [sinα/cosα - sinβ/coβ].
• tan²(θ/2)[sinα.cosβ + cosα.sinβ/cosα.cosβ] = [sinα.cosβ - cosα.sinβ/cosα.cosβ].
• tan²(θ/2) [sinα.cosβ + cosα.sinβ] = [sinα.cosβ - cosα.sinβ]

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Formula of sin (α ± β) = sinα.cosβ ± cosα.sinβ

• tan²(θ/2)[sin(α + β)] = [sin(α - β)].
• tan²(θ/2) = [sin(α - β)]/[sin(α + β)].

☞ tanθ = 1/cot θ

• 1/cot²(θ/2) = [sin(α - β)]/[sin(α + β)].
• [sin(α - β)][cot²(θ/2)] = [sin(α + β)].
• cot²(θ/2) = [sin(α + β)]/[sin(α - β)].

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hope it helps