Class 12 Inverse Trigonometric Functions · Long Answer3 Marksinverse_trigonometric_identities

Class 12 Maths · Inverse Trigonometric Functions · Long Answer #72

For real a,b with b≠0, \(-1\le a/b\le1\) and a≠0, prove \(\tan(\pi/4+\tfrac12\cos^{-1}(a/b))+\tan(\pi/4-\tfrac12\cos^{-1}(a/b))=2b/a\).

✅ Step-by-Step Solution

Final Answer: Identity proved under the stated domain conditions.

Let θ=½arccos(a/b), so θ∈[0,π/2]. The condition a≠0 excludes θ=π/4, the pole of the first tangent.
Use tan U+tan V=sin(U+V)/(cos U cos V), with U=π/4+θ and V=π/4−θ. Here sin(U+V)=1.
The product-to-sum formula gives cos U cos V=[cos2θ+cos(π/2)]/2=cos2θ/2=(a/b)/2.
Thus the sum is 1/[(a/b)/2]=2b/a.

[0.5 marks] Let θ=½arccos(a/b), so θ∈[0,π/2]. The condition a≠0 excludes θ=π/4, the pole of the first tangent.
[1 mark] Use tan U+tan V=sin(U+V)/(cos U cos V), with U=π/4+θ and V=π/4−θ. Here sin(U+V)=1.
[1 mark] The product-to-sum formula gives cos U cos V=[cos2θ+cos(π/2)]/2=cos2θ/2=(a/b)/2.
[0.5 marks] Thus the sum is 1/[(a/b)/2]=2b/a.

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