Question

tan8Q – tan 50 – tan3a = tanga tan5a tan 3a​

Answer 1

$\large\underline\blue{\bold{Given \: Question \: \sf \: (correct \: statement) :- }}$

$\sf \: Prove : tan8x - tan5x - tan3x = tan8x tan5x tan3x$

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$\bf \:\large \red{AηsωeR : } ✍$

$\large\underline\blue{\bold{Formula \: used:- }}$

$\bf \:tan(x + y) = \dfrac{tanx + tany}{1 - tanx \: tany}$

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$\large\underline\purple{\bold{Understanding \: the \: Concept :- }}$

Here the concept of Trigonometric Identities is used. In this question mainly we will use only 1 identity. If we look at the question carefully, there are 3 angles 8x, 5x and 3x such that larger angle 8x represented as a sum of remaining 2 angles 5x and 3x, then we will keep on simplifying and then finally we will get our answer.

Let's do it now!!

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$\large\underline\purple{\bold{Solution :- }}$

$\bf \:Let \: tan8x = tan(5x + 3x)$

$\sf \: ⟼tan8x = \dfrac{tan5x + tan3x}{1 - tan5x \: tan3x}$

☆On cross multiplication, we get

$\sf \: ⟼tan8x - tan8x \: tan5x \: tan3x = tan5x + tan3x$

$\sf \: ⟼tan8x - tan5x - tan3x = tan8xtan5xtan3x$

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$\large \red{\bf \: ⟼ Explore \: more } ✍$

Trigonometry Formulas

• sin(−θ) = −sin θ
• cos(−θ) = cos θ
• tan(−θ) = −tan θ
• cosec(−θ) = −cosecθ
• sec(−θ) = sec θ
• cot(−θ) = −cot θ

Product to Sum Formulas

• sin x sin y = 1/2 [cos(x–y) − cos(x+y)]
• cos x cos y = 1/2[cos(x–y) + cos(x+y)]
• sin x cos y = 1/2[sin(x+y) + sin(x−y)]
• cos x sin y = 1/2[sin(x+y) – sin(x−y)]

Sum to Product Formulas

• sin x + sin y = 2 sin [(x+y)/2] cos [(x-y)/2]
• sin x – sin y = 2 cos [(x+y)/2] sin [(x-y)/2]
• cos x + cos y = 2 cos [(x+y)/2] cos [(x-y)/2]
• cos x – cos y = -2 sin [(x+y)/2] sin [(x-y)/2]

Sum or Difference of angles

• cos (A + B) = cos A cos B – sin A sin B
• cos (A – B) = cos A cos B + sin A sin B
• sin (A+B) = sin A cos B + cos A sin B
• sin (A -B) = sin A cos B – cos A sin B
• tan(A+B) = [(tan A + tan B)/(1 – tan A tan B)]
• tan(A-B) = [(tan A – tan B)/(1 + tan A tan B)]
• cot(A+B) = [(cot A cot B − 1)/(cot B + cot A)]
• cot(A-B) = [(cot A cot B + 1)/(cot B – cot A)]
• cos(A+B) cos(A–B)=cos^2A–sin^2B=cos^2B–sin^2A
• sin(A+B) sin(A–B) = sin^2A–sin^2B=cos^2B–cos^2A

Multiple and Submultiple angles

• sin2A = 2sinA cosA = [2tan A /(1+tan²A)]
• cos2A = cos²A–sin²A = 1–2sin²A = 2cos²A–1= [(1-tan²A)/(1+tan²A)]
• tan 2A = (2 tan A)/(1-tan²A)

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