Find the value of sin37 sin53 tan37 tan53 in terms of fraction

To find the value of sin37° sin53° tan37° tan53°, we can use some trigonometric identities and properties.

First, we know that:

• sin(90° – x) = cos(x),
• Thus, sin53° = cos37°.

Using this, we can rewrite the expression:

sin37° sin53° = sin37° cos37°

Next, we focus on the tangent terms:

• tan(x) = sin(x) / cos(x),
• So, tan37° = sin37° / cos37° and tan53° = sin53° / cos53° = cos37° / sin37°.

Now, multiplying the tangent terms, we have:

tan37° tan53° = (sin37° / cos37°)(cos37° / sin37°) = 1

Now, substituting these values back into our original expression:

sin37° sin53° tan37° tan53° = (sin37° cos37°) * 1 = sin37° cos37°

Finally, applying the identity:

sin(2x) = 2sin(x)cos(x)

Where for x = 37°, we have:

sin(74°) = 2sin37°cos37°

Thus, we can write:

sin37° cos37° = (1/2) sin(74°)

To find sin(74°), as it’s not a special angle, we can find its approximate value or leave it as is. However, since this value involves a known angle, we can express:

sin(74°) = 0.9612616959 (approx)

Hence, the final value is:

sin37° sin53° tan37° tan53° = (1/2) * sin(74°)

In conclusion, the value of sin37° sin53° tan37° tan53° can be expressed as:

(1/2) sin(74°)