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8. Introduction to Trigonometry
medium
Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
$\frac{\cos A}{1+\sin A}+\frac{1+\sin A}{\cos A}=2 \sec A$
Option A
Option B
Option C
Option D
Solution
$\frac{\cos A}{1+\sin A}+\frac{1+\sin A}{\cos A}=2 \sec A$
$L.H.S.\,=\frac{\cos A}{1+\sin A}+\frac{1+\sin A}{\cos A}$
$=\frac{\cos ^{2} A+(1+\sin A)^{2}}{(1+\sin A)(\cos A)}$
$=\frac{\cos ^{2} A+1+\sin ^{2} A+2 \sin A}{(1+\sin A)(\cos A)}$
$=\frac{\sin ^{2} A+\cos ^{2} A+1+2 \sin A}{(1+\sin A)(\cos A)}$
$=\frac{1+1+2 \sin A}{(1+\sin A)(\cos A)}=\frac{2+2 \sin A}{(1+\sin A)(\cos A)}$
$=\frac{2(1+\sin A)}{(1+\sin A)(\cos A)}=\frac{2}{\cos A}=2 \sec A$
$=R . H . S .$
Standard 10
Mathematics
