In a right triangle ABC, right-angled at B, if tanA=1, then verify that 2sinAcosA=1.
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Step-by-Step Solution
Step 1: Define the triangle sides using tan A
Given that tanA=1 in a right-angled triangle ABC, where the right angle is at B. We know that the tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. Therefore, BC (opposite to A) divided by AB (adjacent to A) equals 1.
Step 2: Express sides in terms of a constant
Since ABBC=1, it implies that BC=AB. We can let both sides be equal to a constant k, where k is a positive number, for easier calculation.
Step 3: Calculate the hypotenuse using Pythagoras theorem
Now we use the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (AC) is equal to the sum of the squares of the other two sides (AB and BC). Substituting AB=k and BC=k, we find the length of the hypotenuse AC.
Step 4: Calculate sin A
The sine of angle A is the ratio of the length of the side opposite to angle A (BC) to the length of the hypotenuse (AC). Substituting the values we found, we calculate sinA.
Step 5: Calculate cos A
The cosine of angle A is the ratio of the length of the side adjacent to angle A (AB) to the length of the hypotenuse (AC). Substituting the values we found, we calculate cosA.
Step 6: Verify the expression 2sinAcosA=1
Finally, we substitute the calculated values of sinA and cosA into the given expression 2sinAcosA. We perform the multiplication to show that the expression evaluates to 1, thus verifying the statement. This matches the method shown in NCERT textbook example 4 where tanA=1 is used to verify 2sinAcosA=1.