ত্রিকোমিতি ৯.১ এর ৭(খ) সমাধান।
7) খ) $\displaystyle \frac{secA}{cosA} -\frac{tanA}{cotA} =1$
L.H.S= $\displaystyle \frac{secA}{cosA} -\frac{tanA}{cotA}$
$\displaystyle =\frac{secA}{\frac{1}{secA}} -\frac{tanA}{\frac{1}{tanA}}$
[ We know, $\displaystyle cosA=\frac{1}{secA} \ or,\ cotA=\frac{1}{tanA}$]
$\displaystyle =sec^{2} A-tan^{2} A$ [ we know, $\displaystyle sec^{2} A-tan^{2} A=1$ ]
$\displaystyle =1$
=R.H.S (Proved).
Alternative solution:
L.H.S= $\displaystyle \frac{secA}{cosA} -\frac{tanA}{cotA}$
$\displaystyle =\frac{1}{\frac{cosA}{cosA}} -\frac{1}{\frac{cotA}{cotA}}$
$\displaystyle =\frac{1}{cos^{2} A} -\frac{1}{cot^{2} A}$
$\displaystyle =sec^{2} A-tan^{2} A$
$\displaystyle =1$
=R.H.S
More, Alternative solution:
L.H.S= $\displaystyle \frac{secA}{cosA} -\frac{tanA}{cotA}$
$\displaystyle =\frac{\frac{1}{cosA}}{cosA} -\frac{\frac{sinA}{cosA}}{\frac{cosA}{sinA}}$
[ $\displaystyle secA=\frac{1}{cosA} \ or,\ tnaA=\frac{sinA}{cosA} ,\ cotA=\frac{cosA}{sinA} \ $]
$\displaystyle =\frac{1}{cos^{2} A} -\frac{sin^{2} A}{cos^{2} A}$
$\displaystyle =\frac{1-sin^{2} A}{cos^{2} A}$
$\displaystyle =\frac{cos^{2} A}{cos^{2} A}$
=1
=R.H.S
(Proved).
