Question

1.Evaluate- cos80/sin10+cos59*cosec31 2.If x=a*cos theta-b*sin theta and y=a*sin theta+b*cos theta, then proove that asquare+bsquare=xsquare+ysquare

Answer 1

Hi there
The first question is
cos80/sin10 + cos59 x cosec31
or, cos(90-10)/sin10+cos59/sin31
or,cos(90-10)/sin10+cos(90-31)/sin31
or,sin10/sin10 +sin31/sin31=1+1=2

The second question is
x=acosθ-bsinθ. (1)
and
y=asinθ+bcosθ. (2)
squaring and adding (1) and (2)
we get
x²+y²=(acosθ-bsinθ)²+(asinθ+bcosθ)²
or,
x²+y²=a²cos²θ+b²sin²θ+2absinθcosθ+a²sin²θ
+b²cos²θ-2absinθcosθ
or,x²+y²=a²(sin²θ+cos²θ)+b²(sin²θ+cos²θ)
or,x²+y²=a²+b²
Hope it helps.