Question
Download Solution PDFThe value of: \(\frac{{\sin 23^\circ \cos 67^\circ + \sec52^\circ \sin38^\circ + \cos 23^\circ \sin 67^\circ + \rm cosec52^\circ \cos 38^\circ }}{{\rm cose{c^2}20^\circ - {{\tan }^2}70^\circ }}\)
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFGiven:
The given expression = \(\frac{{\sin 23^\circ \cos 67^\circ + \sec52^\circ \sin38^\circ + \cos 23^\circ \sin 67^\circ + \rm cosec52^\circ \cos 38^\circ }}{{\rm cose{c^2}20^\circ - {{\tan }^2}70^\circ }}\)
Formula used:
cos 67° = sin (90° - 67°) = sin 23°
sin 67° = cos (90° - 67°) = cos 23°
sin2 θ + cos2 θ = 1
sec2 θ - tan2 θ = 1
Calculation:
\(\frac{{\sin 23^\circ \cos 67^\circ + \sec52^\circ \sin38^\circ + \cos 23^\circ \sin 67^\circ + \rm cosec52^\circ \cos 38^\circ }}{{\rm cose{c^2}20^\circ - {{\tan }^2}70^\circ }}\)
= \(\frac{{\sin 23^\circ \sin 23^\circ +\frac{1}{cos\,52^\circ} \sin38^\circ + cos 23^\circ cos 23^\circ + \frac{1}{sin\,52^\circ} \cos 38^\circ }}{sec^2\,70^\circ-tan^2\,70^\circ}\)
= \(\frac{sin^2\,23^\circ+\frac{sin\,38^\circ}{sin\,38^\circ}+cos^2\,23^\circ+\frac{cos\,38^\circ}{cos\,38^\circ}}{1}\)
= sin2 23° + 1 + cos2 23° + 1
= 1 + 1 + 1
= 3
∴ The value of the given expression is 3
Last updated on Jun 13, 2025
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