1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

Prove that: (i) sin 38° + sin 22° = sin 82° (ii) cos 100° + cos 20° = cos 40° (iii) sin 50° + sin 10° = cos 20° (iv) sin 23° + sin 37° = cos 7° (v) sin 105° + cos 105° = cos 45° (vi) sin 40° + sin 20° = cos 10°

Open in App
Solution

(i) $\mathrm{Consider}\mathrm{LHS}:\phantom{\rule{0ex}{0ex}}s\mathrm{in}38°+\mathrm{sin}22°\phantom{\rule{0ex}{0ex}}=2\mathrm{sin}\left(\frac{38°+22°}{2}\right)\mathrm{cos}\left(\frac{38°-22°}{2}\right)\left\{\because \mathrm{sin}A+\mathrm{sin}B=2\mathrm{sin}\left(\frac{A+B}{2}\right)\mathrm{cos}\left(\frac{A-B}{2}\right)\right\}\phantom{\rule{0ex}{0ex}}=2\mathrm{sin}30°\mathrm{cos}8°\phantom{\rule{0ex}{0ex}}=2×\frac{1}{2}\mathrm{cos}\left(90°-8°\right)\phantom{\rule{0ex}{0ex}}=\mathrm{sin}82°\phantom{\rule{0ex}{0ex}}=\mathrm{RHS}\phantom{\rule{0ex}{0ex}}\mathrm{Hence},\mathrm{LHS}=\mathrm{RHS}.$ (ii) $\mathrm{Consider}\mathrm{LHS}:\phantom{\rule{0ex}{0ex}}\mathrm{cos}100°+\mathrm{cos}20°\phantom{\rule{0ex}{0ex}}=2\mathrm{cos}\left(\frac{100°+20°}{2}\right)\mathrm{cos}\left(\frac{100°-20°}{2}\right)\left\{\because \mathrm{cos}A+\mathrm{cos}B=2\mathrm{cos}\left(\frac{A+B}{2}\right)\mathrm{cos}\left(\frac{A-B}{2}\right)\right\}\phantom{\rule{0ex}{0ex}}=2\mathrm{cos}60°\mathrm{cos}40°\phantom{\rule{0ex}{0ex}}=2×\frac{1}{2}\mathrm{cos}40°\phantom{\rule{0ex}{0ex}}=\mathrm{cos}40°\phantom{\rule{0ex}{0ex}}\mathrm{Hence},\mathrm{LHS}=\mathrm{RHS}.$ (iii) $\mathrm{Consider}\mathrm{LHS}:\phantom{\rule{0ex}{0ex}}\mathrm{sin}50°+\mathrm{sin}10°\phantom{\rule{0ex}{0ex}}=2\mathrm{sin}\left(\frac{50°+10°}{2}\right)\mathrm{cos}\left(\frac{50°-10°}{2}\right)\left\{\because \mathrm{sin}A+\mathrm{sin}B=2\mathrm{sin}\left(\frac{A+B}{2}\right)\mathrm{cos}\left(\frac{A-B}{2}\right)\right\}\phantom{\rule{0ex}{0ex}}=2\mathrm{sin}30°\mathrm{cos}20°\phantom{\rule{0ex}{0ex}}=2×\frac{1}{2}\mathrm{cos}20°\phantom{\rule{0ex}{0ex}}=\mathrm{cos}20°\phantom{\rule{0ex}{0ex}}\mathrm{Hence},\mathrm{LHS}=\mathrm{RHS}.$ (iv) $\mathrm{Consider}\mathrm{LHS}:\phantom{\rule{0ex}{0ex}}s\mathrm{in}23°+\mathrm{sin}37°\phantom{\rule{0ex}{0ex}}=2\mathrm{sin}\left(\frac{23°+37°}{2}\right)\mathrm{cos}\left(\frac{23°-37°}{2}\right)\left\{\because \mathrm{sin}A+\mathrm{sin}B=2\mathrm{si}n\left(\frac{A+B}{2}\right)\mathrm{cos}\left(\frac{A-B}{2}\right)\right\}\phantom{\rule{0ex}{0ex}}=2\mathrm{sin}30°\mathrm{cos}\left(-7°\right)\phantom{\rule{0ex}{0ex}}=2\mathrm{sin}30°\mathrm{cos}7°\phantom{\rule{0ex}{0ex}}=2×\frac{1}{2}\mathrm{cos}7°\phantom{\rule{0ex}{0ex}}=\mathrm{cos}7°\phantom{\rule{0ex}{0ex}}\mathrm{Hence},\mathrm{LHS}=\mathrm{RHS}.$ (v) $\mathrm{Consider}\mathrm{LHS}:\phantom{\rule{0ex}{0ex}}s\mathrm{in}105°+\mathrm{cos}105°\phantom{\rule{0ex}{0ex}}=\mathrm{sin}105°+\mathrm{cos}\left(90°+15°\right)\phantom{\rule{0ex}{0ex}}=\mathrm{sin}105°-\mathrm{sin}15°\phantom{\rule{0ex}{0ex}}=2\mathrm{si}n\left(\frac{105°-15°}{2}\right)\mathrm{cos}\left(\frac{105°+15°}{2}\right)\left\{\because \mathrm{sin}A+\mathrm{sin}B=2\mathrm{sin}\left(\frac{A-B}{2}\right)\mathrm{cos}\left(\frac{A+B}{2}\right)\right\}\phantom{\rule{0ex}{0ex}}=2\mathrm{sin}45°\mathrm{cos}60°\phantom{\rule{0ex}{0ex}}=2\mathrm{sin}\left(90°-45°\right)\mathrm{cos}60°\phantom{\rule{0ex}{0ex}}=2×\frac{1}{2}\mathrm{cos}\left(45°\right)\phantom{\rule{0ex}{0ex}}=\mathrm{cos}45°\phantom{\rule{0ex}{0ex}}\mathrm{Hence},\mathrm{LHS}=\mathrm{RHS}.$ (vi) $\mathrm{Consider}\mathrm{LHS}:\phantom{\rule{0ex}{0ex}}\mathrm{sin}40°+\mathrm{sin}20°\phantom{\rule{0ex}{0ex}}=2\mathrm{sin}\left(\frac{40°+20°}{2}\right)\mathrm{cos}\left(\frac{40°-20°}{2}\right)\left\{\because \mathrm{sin}A+\mathrm{sin}B=2\mathrm{sin}\left(\frac{A+B}{2}\right)\mathrm{cos}\left(\frac{A-B}{2}\right)\right\}\phantom{\rule{0ex}{0ex}}=2\mathrm{sin}30°\mathrm{cos}10°\phantom{\rule{0ex}{0ex}}=2×\frac{1}{2}\mathrm{cos}10°\phantom{\rule{0ex}{0ex}}=\mathrm{cos}\left(10°\right)\phantom{\rule{0ex}{0ex}}\mathrm{Hence},\mathrm{LHS}=\mathrm{RHS}.$

Suggest Corrections
0
Join BYJU'S Learning Program
Related Videos
Trigonometric Ratios of Allied Angles
MATHEMATICS
Watch in App
Join BYJU'S Learning Program