MHT CET · Maths · Trigonometric Equations
If \(\sec x+\tan x=3, x \in\left(0, \frac{\pi}{2}\right)\) then, \(\sin x=\)
- A \(\frac{3}{5}\)
- B \(\frac{4}{5}\)
- C \(-1\)
- D \(\frac{1}{5}\)
Answer & Solution
Correct Answer
(B) \(\frac{4}{5}\)
Step-by-step Solution
Detailed explanation
We have \(\sec x+\tan x=3\) ...(1)
We know that \(\sec ^{2} x-\tan ^{2} x=1\)
\(\therefore(\sec x-\tan x)(\sec x+\tan x)=1 \Rightarrow \sec x-\tan x=\frac{1}{3}\) ...(2)
Adding (1) and (2)
\(2 \sec x=\frac{10}{3} \Rightarrow \sec x=\frac{5}{3} \Rightarrow \cos x=\frac{3}{5} \Rightarrow \sin x=\sqrt{1-\cos ^{2} x}=\frac{4}{5}\)
We know that \(\sec ^{2} x-\tan ^{2} x=1\)
\(\therefore(\sec x-\tan x)(\sec x+\tan x)=1 \Rightarrow \sec x-\tan x=\frac{1}{3}\) ...(2)
Adding (1) and (2)
\(2 \sec x=\frac{10}{3} \Rightarrow \sec x=\frac{5}{3} \Rightarrow \cos x=\frac{3}{5} \Rightarrow \sin x=\sqrt{1-\cos ^{2} x}=\frac{4}{5}\)
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