Answer:
The answer is 90 (45+45)
The procedure is given in the above picture
Hope it helps you
If (1 + tanθ)(1 + tanΦ) = 2.
As we know that,
Formula of :
⇒ tan(x + y) = [(tan x + tan y)/(1 - tan x tan y)].
Using this formula in this question, we get.
⇒ (1 + tanθ)(1 + tanΦ) = 2.
⇒ 1 x (1 + tanΦ) + tanθ x (1 + tanΦ) = 2.
⇒ 1 + tanΦ + tanθ + tanθ tanΦ = 2.
⇒ tanθ + tanΦ = 2 - 1 - tanθ tanΦ.
⇒ tanθ + tanΦ = 1 - tanθ tanΦ.
⇒ (tanθ + tanΦ)/(1 - tanθ tanΦ) = 1.
⇒ tan(θ + Φ) = 1.
⇒ tan(θ + Φ) = tan(45°).
⇒ (θ + Φ) = 45°
Option [B] is correct answer.
Trigonometric ratios of multiple angles.
sin2θ = 2sinθcosθ = 2tanθ/(1 + tan²θ).
cos2θ = 2cos²θ - 1 = 1 - 2sin²θ = cos²θ - sin²θ = (1 - tan²θ)/(1 + tan²θ).
tan2θ = 2tanθ/(1 - tan²θ).
sin3θ = 3sinθ - 4sin³θ.
cos3θ = 4cos³θ - 3cosθ.
tan3θ = (3tanθ - tan³θ)/(1 - 3tan²θ).
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Answers & Comments
Answer:
The answer is 90 (45+45)
The procedure is given in the above picture
Hope it helps you
Verified answer
EXPLANATION.
If (1 + tanθ)(1 + tanΦ) = 2.
As we know that,
Formula of :
⇒ tan(x + y) = [(tan x + tan y)/(1 - tan x tan y)].
Using this formula in this question, we get.
⇒ (1 + tanθ)(1 + tanΦ) = 2.
⇒ 1 x (1 + tanΦ) + tanθ x (1 + tanΦ) = 2.
⇒ 1 + tanΦ + tanθ + tanθ tanΦ = 2.
⇒ tanθ + tanΦ = 2 - 1 - tanθ tanΦ.
⇒ tanθ + tanΦ = 1 - tanθ tanΦ.
⇒ (tanθ + tanΦ)/(1 - tanθ tanΦ) = 1.
⇒ tan(θ + Φ) = 1.
⇒ tan(θ + Φ) = tan(45°).
⇒ (θ + Φ) = 45°
Option [B] is correct answer.
MORE INFORMATION.
Trigonometric ratios of multiple angles.
sin2θ = 2sinθcosθ = 2tanθ/(1 + tan²θ).
cos2θ = 2cos²θ - 1 = 1 - 2sin²θ = cos²θ - sin²θ = (1 - tan²θ)/(1 + tan²θ).
tan2θ = 2tanθ/(1 - tan²θ).
sin3θ = 3sinθ - 4sin³θ.
cos3θ = 4cos³θ - 3cosθ.
tan3θ = (3tanθ - tan³θ)/(1 - 3tan²θ).