Question

Sin alpha + cos alpha=1 then prove that sin alpha - cos alpha =+-1

Answer 1

REFER TO ATTACHMENT:.....

GIVEN:

$sin\alpha+cos\alpha=1$

TO PROVE:

$sin\alpha-cos\alpha=±1$

ANSWER:

We have,

=>$sin\alpha+cos\alpha=1$

Squaring both sides,

=>$(sin\alpha+cos\alpha) ^{2}=1^{2}$

=>$sin^{2}\alpha+cos^{2}\alpha+2sin\alpha cos\alpha=1$

$\large\red{\boxed{(a+b) ^{2}=a^{2}+b^{2}+2ab}}$

=>$1+2sin\alpha cos\alpha=1$

$\large\green{\boxed{sin^{2}\theta+cos^{2}\theta=1}}$

=>$2sin\alpha cos\alpha=1-1$

=>$2sin\alpha cos\alpha=0$ ......... (1)

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Let $sin\alpha-cos\alpha=y$

On squaring both sides,

=>$(sin\alpha-cos\alpha) ^{2}=y^{2}$

=>$sin^{2}\alpha+cos^{2}\alpha-2sin\alpha cos\alpha= y^{2}$

$\large\purple{\boxed{(a-b) ^{2}=a^{2}+b^{2}-2ab}}$

$\large\blue{\boxed{sin^{2}\theta+cos^{2}\theta=1}}$

=>$1-0=y^{2}$

(from ...... (1))

=>$y^{2}=1$

=>$y=\sqrt{1}$

. °. $y =±1$

We had earlier assumed that $sin\alpha-cos\alpha=y$

Hence proved

$\huge\orange{\boxed{sin\alpha-cos\alpha=±1}}$

Answer 2

QUESTION :

$\sf{If\:sin\alpha+cos\alpha=1}$

$\sf{Then\: prove\: that\:sin\alpha-cos\alpha=\frac{+}{}1}$

SOLUTION :

$\sf{sin\alpha+cos\alpha=1}$

✪We know the identity : sin²A+ cos²A = 1✪

So,

$\sf{{sin}^{2}\alpha+{cos}^{2}\alpha=1}$

✪Now use the identity : a²+b²= (a+b)²-2ab✪

$\implies\sf{(sin\alpha+cos\alpha)^2-2sin\alpha.cos\alpha=1}$

✪Put the value of sinα + cosα=1✪

$\implies\sf{(1)^2-2sin\alpha.cos\alpha=1}$

$\implies\sf{1-2sin\alpha.cos\alpha=1}$

$\implies\sf{-2sin\alpha.cos\alpha=1-1}$

$\implies\sf{2sin\alpha.cos\alpha=0}$

Again,

$\sf{{sin}^{2}\alpha+{cos}^{2}\alpha=1}$

✪ Now use the identity : a²+b²= (a-b) ²+2ab ✪

$\implies\sf{(sin\alpha-cos\alpha)^2+2sin\alpha.cos\alpha=1}$

✪Put the value of 2sinα.cosα = 0✪

$\implies\sf{(sin\alpha-cos\alpha)^2-0=1}$

$\implies\sf{(sin\alpha-cos\alpha)^2=1+0}$

$\implies\sf{(sin\alpha-cos\alpha)^2=1}$

$\implies\sf{(sin\alpha-cos\alpha)=\sqrt{1}}$

$\implies\sf{(sin\alpha-cos\alpha)=\frac{+}{}1}$

Hence proved_____!!

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MorE InformatioN :

Some identities related to trigonometry :-

★ 1+ tan²A = sec²A

★ 1+cot²A = cosec²A

★ cos²A - sin²A = cos2A

★ sin2A = 2sinA.cosA

★ cos2A = 1- sin²A

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