Show that : (i) tan48^∘ tan23^∘ tan42^∘ tan67^∘=1 (ii)...

Key Concepts

Complementary Angle Relationship

This concept involves pairs of angles whose measures add up to 90°. In trigonometry, complementary angles have interrelated sine, cosine, and tangent values, such that the sine of one angle equals the cosine of its complement and the tangent of one angle equals the cotangent of its complement. This property often simplifies problems involving products or sums of trigonometric functions.

Tangent-Cotangent Identity

The tangent of an angle and the tangent of its complement are reciprocals of each other; specifically, tan(x) equals the cotangent of (90° ? x). This immediately leads to the identity that the product of tan(x) with tan(90° ? x) is 1, a powerful tool in solving trigonometric products involving pairs of complementary angles.

Cosine Addition Formula

This formula expresses the cosine of the sum of two angles in terms of the cosine and sine of the individual angles: cos(A + B) = cos A cos B ? sin A sin B. It is a fundamental identity in trigonometry, enabling the conversion of sums or differences into products or single function evaluations, which can simplify calculations and proofs.

Special Angle Trigonometric Values

Certain angles, such as 90° or its complementary angles, produce trigonometric function values that are especially simple (for example, cos 90° equals 0). Knowing these values and how they arise from addition formulas or complementary identities is important for verifying trigonometric identities and simplifying expressions.

Transcript

00:01 So in the given question we are told to show that first we are told to show that tan of 48 degrees times tan of 23 degrees times tan of 42 degrees times tan of 67 degrees is equal to 1.

00:25 So, the identity that we can use over here is we can use that tan theta can be written as cot of 90 minus theta and we are also going to use the identity that tan theta is equal to 1 by cot theta right so then let's write tan of 48 degrees instead of tan of 23 degrees let's write cot of 90 minus 23 degrees and instead of tan 42 degrees that's right caught of 90 minus 42 degrees times tan of 67 so this would then be equal to we have tan of 48 degrees times cot of 90 minus 23 is 67 degrees then we have got of 90 which is 48 degrees times tan of 67 degrees, right? so now what we can do is we have tan 48 and cot 48, right? so now let's use this identity that tan theta is equal to 1 by cot 48 which means we can write tan theta as 1 by cot 48 times we have got 48 and times we have got 67 and and tan 67 can be written as cot of 1 by cot of 67 which means we can divide cot 48 by cot 48 and cot 67 by cot 61 67 giving us the value as 1.

02:11 So this is how we show that the given expression is equal to 1.

02:15 So there is a second expression that is given in the question which is given as cost of 38, cost of 38 degrees times cost of 52 degrees minus sine of 38 degrees times sine of 52 degrees is equal to 0.

02:44 So we need to show that this expression would evaluate to give us the result as 0.

02:50 So the identity that we can use over here is that cost of an angle theta can be written as sign of 90 minus theta.

03:03 So using this property, let's first write instead of cost 52, we can write cost 38 degrees times.

03:15 Instead of cost 52 degrees, we can write sign of 90 degrees.

03:21 Minus 52 and this minus sign of 38 degrees times sign of 52 degrees...

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