Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
$(\sin A+\operatorname{cosec} A)^{2}+(\cos A+\sec A)^{2}=7+\tan ^{2} A+\cot ^{2} A$
Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
$(\sin A+\operatorname{cosec} A)^{2}+(\cos A+\sec A)^{2}=7+\tan ^{2} A+\cot ^{2} A$
$(\sin A+\operatorname{cosec} A)^{2}+(\cos A+\sec A)^{2}=7+\tan ^{2} A+\cot ^{2} A$
$L.H.S.=(\sin A+\operatorname{cosec} A)^{2}+(\cos A+\sec A)^{2}$
$\quad=\sin ^{2} A+\operatorname{cosec}^{2} A+2 \sin A \operatorname{cosec} A+\cos ^{2} A+\sec ^{2} A+2 \cos A \sec A$
$\quad=\left(\sin ^{2} A+\cos ^{2} A\right)+\left(\operatorname{cosec}^{2} A+\sec ^{2} A\right)+2 \sin A\left(\frac{1}{\sin A}\right)+2 \cos A\left(\frac{1}{\cos A}\right)$
$\quad=(1)+\left(1+\cot ^{2} A+1+\tan ^{2} A\right)+(2)+(2)$
$\quad=7+\tan ^{2} A+\cot ^{2} A$
$=R \cdot H . S.$
Similar Questions
In $\triangle PQR ,$ right $-$ angled at $Q , PR + QR =25\, cm$ and $PQ =5\, cm .$ Determine the values of $\sin P, \cos P$ and $\tan P$.
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In $Fig.$ find $\tan P-\cot R .$
In $Fig.$ find $\tan P-\cot R .$
In triangle $ABC ,$ right -angled at $B ,$ if $\tan A =\frac{1}{\sqrt{3}},$ find the value of:
$(i)$ $\sin A \cos C+\cos A \sin C$
$(ii)$ $\cos A \cos C-\sin A \sin C$
In triangle $ABC ,$ right -angled at $B ,$ if $\tan A =\frac{1}{\sqrt{3}},$ find the value of:
$(i)$ $\sin A \cos C+\cos A \sin C$
$(ii)$ $\cos A \cos C-\sin A \sin C$
$(1+\tan \theta+\sec \theta)(1+\cot \theta-\operatorname{cosec} \theta)=..........$
$(1+\tan \theta+\sec \theta)(1+\cot \theta-\operatorname{cosec} \theta)=..........$
Evaluate the following:
$\sin 60^{\circ} \cos 30^{\circ}+\sin 30^{\circ} \cos 60^{\circ}$
Evaluate the following:
$\sin 60^{\circ} \cos 30^{\circ}+\sin 30^{\circ} \cos 60^{\circ}$