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If \(4 \cos x+3 \sin x=5\), then find the value of \(\tan x=\)
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Verified Answer
The correct answer is:
\(\frac{3}{4}\)
Given, \(4 \cos x+3 \sin x=5\)
\(\Rightarrow \quad 4+3 \tan x=5 \sec x\)
On squaring both sides, we get
\(\begin{array}{rlrl}
& 16+9 \tan ^2 x+24 \tan x & =25 \sec ^2 x \\
\Rightarrow \quad & 16 \tan ^2 x-24 \tan x+9 & =0 \\
\Rightarrow \quad(4 \tan x-3)^2=0 \Rightarrow \tan x & =\frac{3}{4}
\end{array}\)
\(\Rightarrow \quad 4+3 \tan x=5 \sec x\)
On squaring both sides, we get
\(\begin{array}{rlrl}
& 16+9 \tan ^2 x+24 \tan x & =25 \sec ^2 x \\
\Rightarrow \quad & 16 \tan ^2 x-24 \tan x+9 & =0 \\
\Rightarrow \quad(4 \tan x-3)^2=0 \Rightarrow \tan x & =\frac{3}{4}
\end{array}\)
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