Question

If cos theta-sin theta = 1. then theta=
1)π
2)3π/2
3)2π/3
4)π/6​

Answer 1

Question :-

$\sf{if \: \cos \theta \: - \sin \theta = 1 \: then \: find \: \theta \: .}$

Answer :-

$\boxed{ \implies} \: \boxed{\theta \: = \frac{ \pi}{2} } \:$

To Find :-

$\boxed{ \star} \: \: \sf{find \: the \: value \: of \: \theta} \: .$

Formula used :-

$\implies \: { \sin}^{2} \theta \: + { \cos}^{2} \theta \: = 1 \\ \\ \implies \sf{ {(x - y)}^{2} = {x}^{2} + {y}^{2} - 2xy} \\ \\ \implies \: \sin 2 \theta \: = 2 \sin \theta \: cos \theta$

Explanation :-

According to the question,

$\rightarrow \: \cos \theta \: - \sin \theta \: = 1 \\ \\ \text{squaring \: on \: both \: sides} \\ \\ \rightarrow \: {( \cos \theta - \sin \theta)}^{2} = {1}^{2} \\ \\ \rightarrow \: { \cos}^{2} \theta \: + { \sin}^{2} \theta \: - 2 \cos \theta \: \sin \theta = 1 \\ \\ \rightarrow \: 1 - \sin 2 \theta \: = 1 \\ \\ \rightarrow \: \sin 2 \theta = 0 \\ \\ \rightarrow \: \sin \: 2 \theta \: = \sin \pi \\ \\ \rightarrow \: 2 \theta \: = \pi \\ \\ \rightarrow \: \boxed{\theta \: = \frac{ \pi}{2} }$

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