EXAMPLE 28 Prove that tan cOS X (1 ...
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EXAMPLE 28 Prove that tan cOS X (1 + sin x

11th - 12th Class
Maths
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143
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soLUTION We have [ begin{aligned} L H S &=tan ^{-1}left(frac{cos x}{1+sin x}right) &=tan ^{-1}left{frac{sin left(frac{pi}{2}-xright)}{1+cos left(frac{pi}{2}-xright)}right} &=tan ^{-1}left{frac{2 sin left(frac{pi}{4}-frac{x}{2}right) cos left(frac{pi}{4}-frac{x}{2}right)}{2 cos ^{2}left(frac{pi}{4}-frac{x}{2}right)}right} therefore tan ^{-1}left(frac{cos x}{1+sin x}right)=left(frac{pi}{4}-frac{x}{2}right) & text { tan }left.left(frac{pi}{4}-frac{x}{2}right)right}=left(frac{pi}{4}-frac{x}{2}right)=text { RHS. } end{aligned} ]
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