CBSE 2025 · Region 6 · Set 1 · Q22 · 2 marks
If a $\displaystyle \sec \theta+\mathrm{b} \tan \theta=\mathrm{m}$ and $\displaystyle \mathrm{b} \sec \theta+\mathrm{a} \tan \theta=\mathrm{n}$, prove that $\displaystyle \mathrm{a}^{2}+\mathrm{n}^{2}=\mathrm{b}^{2}+\mathrm{m}^{2}$Use the identity : $\displaystyle \sin ^{2} \mathrm{~A}+\cos ^{2} \mathrm{~A}=1$ to prove that $\displaystyle \tan ^{2} \mathrm{~A}+1=\sec ^{2} \mathrm{~A}$. Hence, find the value of $\displaystyle \tan \mathrm{A}$, when $\displaystyle \sec \mathrm{A}=\frac{5}{3}$, where A is an acute angle.
If a $\displaystyle \sec \theta+\mathrm{b} \tan \theta=\mathrm{m}$ and $\displaystyle \mathrm{b} \sec \theta+\mathrm{a} \tan \theta=\mathrm{n}$, prove that $\displaystyle \mathrm{a}^{2}+\mathrm{n}^{2}=\mathrm{b}^{2}+\mathrm{m}^{2}$
Use the identity : $\displaystyle \sin ^{2} \mathrm{~A}+\cos ^{2} \mathrm{~A}=1$ to prove that $\displaystyle \tan ^{2} \mathrm{~A}+1=\sec ^{2} \mathrm{~A}$. Hence, find the value of $\displaystyle \tan \mathrm{A}$, when $\displaystyle \sec \mathrm{A}=\frac{5}{3}$, where A is an acute angle.
Marking-scheme solution
\[m^{2}=a^{2} \sec ^{2} \theta+b^{2} \tan ^{2} \theta+2 a b \sec \theta \tan \theta
\]
\(\displaystyle \mathrm{n}^{2}=\mathrm{b}^{2} \sec ^{2} \theta+\mathrm{a}^{2} \tan ^{2} \theta+2 \mathrm{ab} \sec \theta \tan \theta\)
\[m^{2}-n^{2}=a^{2}\left(\sec ^{2} \theta-\tan ^{2} \theta\right)+b^{2}\left(\tan ^{2} \theta-\sec ^{2} \theta\right)
\]
\(\displaystyle \Rightarrow \mathrm{m}^{2}-\mathrm{n}^{2}=\mathrm{a}^{2}-\mathrm{b}^{2}\) or \(\displaystyle \mathrm{a}^{2}+\mathrm{n}^{2}=\mathrm{m}^{2}+\mathrm{b}^{2}\)
\(\displaystyle \sin ^{2} \mathrm{~A}+\cos ^{2} \mathrm{~A}=1\)
Dividing both sides by \(\displaystyle \cos ^{2} \mathrm{~A}\), we get
\[\frac{\sin ^{2} A}{\cos ^{2} A}+\frac{\cos ^{2} A}{\cos ^{2} A}=\frac{1}{\cos ^{2} A}
\]
\(\displaystyle \tan ^{2} \mathrm{~A}+1=\sec ^{2} \mathrm{~A}\)
\(\displaystyle \tan ^{2} \mathrm{~A}+1=\left(\frac{5}{3}\right)^{2}\)
\(\displaystyle \tan \mathrm{A}=\frac{4}{3}\)
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CBSE Class 10 Mathematics past-paper question from the 2025board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.