Question

If m = tan ɵ +sin ɵ and n= tan ɵ - sin ɵ ,show that m2 – n2 = 4√mn​

Answer 1

Step-by-step explanation:

$\huge\begin{lgathered}\frak\red{answEr}\end{lgathered}$

$\tt assuming\:theta\:to\:be\:a$

$\huge\mathfrak\blue{given::}$

$\tt m\:=\:tana + sina$

$\tt n\:=\:tana-sina$

$\huge\mathfrak\green{to\:prove::}$

$\tt m^2\:-\:n^2\:=\:4✓mn$

we know that,

$\tt (tana + Sina)^2\:-\:(tana-sina)^2=$

$\tt tan^2a + sin^2 a + 2tanasina - tan^2a$

$\tt - sin^2a + 2tanasina$

====>$\tt 4tanasina$

$\tt also,$

$\tt m=tana + Sina$

$\tt n = tana-sina$

replacing in above equation,,

$\tt m^2-n^2= 4tanasina$.......l.h.s

$\tt 4✓mn = R.H.S.$

=====>$\tt 4×✓(tan^2a-sin^2a)$

=====> $\tt 4×✓(\dfrac{sin^2a}{cos^2a} - sin^2a)$

=====> $\tt taking\:lcm,$

=====> $\tt 4 × ✓ (\dfrac{sin^2a-sin^2acos^2a}{cos^2a})$

=====>$\tt 4 × ✓\dfrac{sin^2a(1-cos^2a)}{cos^2a}$

=====>$\tt 4 × Sina ✓\dfrac {sin^2a}{cos^2a}$

=====> $\tt 4× sina×tana$

since,

$\tt L.H.S=R.H.S = 4sinatana$

$\tt the\:question\:gets\:proved$

$\huge\begin{lgathered}\frak\red{hoPe\:iT\:helPs}\end{lgathered}$

Answer 2

Answer:

m=tanɵ +sinɵ and n= tanɵ - sinɵ

m2-n2=(tanɵ +sinɵ)2-(tanɵ - sinɵ)2

=2tanɵ+2sinɵ-2tanɵ+2sinɵ

=4sinɵ

Again, m = tanɵ+sinɵ=>sinɵ=m- tanɵ••••••1

n=tanɵ - sinɵ=>sinɵ=tanɵ-n••••••••2

From 1 & 2=>

m- tanɵ=tanɵ-n

=>2tanɵ=m+n

=>tanɵ=m+n/2•••••••••••3

Now, putting the value of tanɵ in eq.n 1

sinɵ=m-m+n/2

=m+n/2

So,4sinɵ=4(m+n/2)

=2(m+n)

May be the Question is send wrongly.