Class 12 Inverse Trigonometric Functions · Long Answer3 Marksbranch_sensitive_simplification

Class 12 Maths · Inverse Trigonometric Functions · Long Answer #94

For \(\pi<x<3\pi/2\), simplify \(\tan^{-1}\frac{\sqrt{1+\cos x}+\sqrt{1-\cos x}}{\sqrt{1+\cos x}-\sqrt{1-\cos x}}\).

✅ Step-by-Step Solution

Final Answer: π/4−x/2

For π<x<3π/2, put t=x/2∈(π/2,3π/4). Then cos t<0 and sin t>0.
Hence √(1+cos x)=−√2 cos t and √(1−cos x)=√2 sin t. Their sum divided by their difference becomes (cos t−sin t)/(cos t+sin t).
This quotient is tan(π/4−t). Also π/4−t∈(−π/2,−π/4), within the principal arctangent interval.
Therefore the expression is π/4−x/2.

[0.5 marks] For π<x<3π/2, put t=x/2∈(π/2,3π/4). Then cos t<0 and sin t>0.
[1 mark] Hence √(1+cos x)=−√2 cos t and √(1−cos x)=√2 sin t. Their sum divided by their difference becomes (cos t−sin t)/(cos t+sin t).
[1 mark] This quotient is tan(π/4−t). Also π/4−t∈(−π/2,−π/4), within the principal arctangent interval.
[0.5 marks] Therefore the expression is π/4−x/2.

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