Question

$cos^{2} \alpha - \sin^{2} \alpha = \tan^{2} \beta \\ prove \: that \: \cos\beta = 1 \div \sqrt{2} \cos \alpha$

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Answer 1

$\large\underline\blue{\bold{Given \:  Question :-  }}$

$\tt \:  cos^{2} \alpha - \sin^{2} \alpha = \tan^{2} \beta \: \: \: \: \: \: \: \: \\ \tt \:  ⟼ prove \: that \: \cos\beta = 1 \div \sqrt{2} \cos \alpha </p><p>$

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$\begin{gathered}\Large{\bold{\pink{\underline{Formula \: Used \::}}}} \end{gathered}$

$\large \boxed{ \tt \:  \red{ ⟼ (1) \:cos2x = {cos}^{2} x - {sin}^{2} x}}$

$\large \boxed{ \tt \:  \red{ ⟼ (2) \: 1 + cos2x = 2 {cos}^{2} x}}$

$\large \boxed{ \tt \:  \red{ ⟼ (3) \: {sec}^{2} x - {tan}^{2} x = 1}}$

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

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$\tt \:  cos^{2} \alpha - \sin^{2} \alpha = \tan^{2} \beta$

$\tt : \implies \:cos2 \alpha = {sec}^{2} \ \beta - 1$

$\tt : \implies \: {sec}^{2} \beta = 1 + cos2 \alpha$

$\tt : \implies \: {2cos}^{2} \alpha = {sec}^{2} \beta$

$\tt : \implies \:sec \beta = \sqrt{2} cos \alpha$

$\tt : \implies \:\dfrac{1}{cos \beta } = \sqrt{2} cos \alpha$

$\tt : \implies \:cos \beta = \dfrac{1}{ \sqrt{2}cos \alpha }$

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$\large{\boxed{\boxed{\bf{Hence, Proved}}}}$

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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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