Jambmaths question:
If $\tan \theta =\tfrac{4}{3}$, calculate ${{\sin }^{2}}\theta -{{\cos }^{2}}\theta $
Option A:
$\tfrac{16}{25}$
Option B:
$\tfrac{24}{25}$
Option C:
$\tfrac{7}{25}$
Option D:
$\tfrac{9}{25}$
Jamb Maths Solution:
$\begin{align} & \text{From the definition of tangent of an angle} \\ & \tan \theta =\frac{opposite}{adjacent}=\frac{4}{3} \\ & A{{C}^{2}}={{3}^{3}}+{{4}^{2}}\text{ (Pythagoras theorem)} \\ & AC=5 \\ & \cos \theta =\tfrac{adj}{hyp}=\tfrac{3}{5},\text{ }{{\cos }^{2}}\theta ={{(\tfrac{3}{5})}^{2}}=\tfrac{9}{25} \\ & \sin \theta =\tfrac{opp}{hyp}=\tfrac{4}{5},\text{ }{{\sin }^{2}}\theta ={{(\tfrac{4}{5})}^{2}}=\tfrac{16}{25} \\ & {{\sin }^{2}}\theta -{{\cos }^{2}}\theta =\tfrac{16}{25}-\tfrac{9}{25}=\tfrac{7}{25} \\\end{align}$
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