ত্রিকোমিতি ৯.১ এর ৭(ক) সমাধান।
7) ক) $\displaystyle \frac{sinA}{cosecA} +\frac{cosA}{secA} =1$
L.H.S = $\displaystyle \frac{sinA}{cosecA} +\frac{cosA}{secA}$
$\displaystyle =\frac{sinA}{\frac{1}{sinA}} +\frac{cosA}{\frac{1}{cosA}}$
[ We know, $\displaystyle cosecA=\frac{1}{sinA} \ or,\ secA=\frac{1}{cosA}$ ]
$\displaystyle =sin^{2} A+cos^{2} A$ [ We know, $\displaystyle sin^{2} A+cos^{2} A$ ]
$\displaystyle =1$
=R.H.S
Alternative solution:
L.H.S = $\displaystyle \frac{sinA}{cosecA} +\frac{cosA}{secA}$
$\displaystyle =\frac{\frac{1}{cosecA}}{cosecA} +\frac{\frac{1}{secA}}{secA}$
[ We know, $\displaystyle sinA=\frac{1}{cosecA} \ or,\ cosA=\frac{1}{secA}$]
$\displaystyle =\frac{1}{cosec^{2} A} +\frac{1}{sec^{2} A}$
$\displaystyle =\frac{1}{\frac{1}{sin^{2} A}} +\frac{1}{\frac{1}{cos^{2} A}}$
$\displaystyle =sin^{2} A+cos^{2} A$
[ we know, $\displaystyle sin^{2} A+cos^{2} A=1$ ]
$\displaystyle =1$
= R.H.S
so, $\displaystyle \frac{sinA}{cosecA} +\frac{cosA}{secA} =1$ ( Proved)
