This CBSE class 10 Maths additional practice question is from the topic Trigonometry. Concepts Covered: Trigonometric Values, Pythagoras Theorem, and Similarity of Triangles.

Question 3 : If C and Z are acute angles and that cos C = cos Z prove that ∠C = ∠Z .

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**Given** C and Z are acute angles and cos C = cos Z

cos C = \\frac{\text{BC}}{\text{AC}}) and cos Z = \\frac{\text{YZ}}{\text{XZ}})

cos C = cos Z =>\\frac{\text{BC}}{\text{AC}})= \\frac{\text{YZ}}{\text{XZ}})

Let \\frac{\text{BC}}{\text{YZ}})= \\frac{\text{AC}}{\text{XZ}}) = k**..... (1)**

**BC = k(yz) and AC = k(xz)**

In right triangle ABC, using Pythagoras theorem AB = \\sqrt{AC^2− BC^2 })

AB = \\sqrt{(k(xz)^2−(k(yz)^2)})= k\\sqrt{(xz)^2− (yz)^2 })

In right triangle xyz, using Pythagoras theorem xy = \\sqrt{(xz)^2− (yz)^2 })

\\frac{\text{AB}}{\text{XY}}) = \\frac{K × \sqrt{(X Z)^{2}-(Y Z)^{2}}}{\sqrt{(X Z)^{2}-(Y Z)^{2}}})

Or \\frac{\text{AB}}{\text{XY}}) = k **..... (1)**

From **1** and **2**, \\frac{\text{BC}}{\text{YZ}}) = \\frac{\text{AC}}{\text{XZ}}) = \\frac{\text{AB}}{\text{XY}}) = k

By SSS similarity criterion, we can conclude ΔABC is similar to ΔXYZ

∴ Corresponding angles of two similar triangles will be equal.

∴ **∠C = ∠Z **

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CBSE Class 10 Maths - 2021

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