Question

if (a. a-b. b)sinx+2absosx=a. a+b. b find the value of tanx​

Answer 1

you mean, (a² - b²) sinx + 2abcosx = a² + b²

then we have to find value of tanx.

(a² - b²)sinx + 2abcosx = (a² + b²)

or, (a² - b²)sinx/cosx + 2abcosx/cosx = (a² + b²)/cosx

or, (a² - b²)tanx + 2ab = (a² + b²)secx

squaring both sides,

or, (a² - b²)²tan²x + 4a²b² + 2ab(a² - b²)tanx = (a² + b²)²sec²x

or, (a² - b²)²tan²x + 4a²b² + 2ab(a² - b²)tanx = (a² + b²)²(1 + tan²x) [ as you know as well, sec²x - tan²x = 1]

or, (a² - b²)²tan²x + 2ab(a² - b²)tanx -(a² + b²)²tan²x + 4a²b² - (a² + b²)² = 0

or, (a⁴ + b⁴ - 2a²b² - a⁴ - b⁴ - 2a²b²)tan²x + 2ab(a² - b²)tanx + 4a²b² -(a⁴ + b⁴ + 2a²b²) = 0

or, (-4a²b²)tan²x + 2ab(a² - b²)tanx -(a² - b²)² = 0

or, (2ab)²tan²x + 2ab(a² - b²)tanx + (a² - b²)² = 0

or, [(2ab)tanx - (a² - b²)]² = 0

or, (2ab)tanx = (a² - b²)

or, tanx = (a² - b²)/2ab