If A = cos2 theta + sin4 theta, then prove that for all the values of theta, 3/4 equal - Maths - Trigonometric Functions

Question

If A = cos2 theta + sin4 theta, then prove
that for all the values of theta, 3/4< equal to A < equal to
1

Anuradha Sharma · Accepted Answer

I think your question is, A = cos2θ+sin4θ=1× cos2θ+sin4θ=  cos2θ cos2θ+sin2θ+sin4θ  Because 1 =cos2θ+sin2θ =cos4θ+sin2θcos2θ+sin4θNow adding and subtracting sin2θcos2θ= cos4θ+2sin2θcos2θ+sin4θ-sin2θcos2= cos2θ+sin2θ2-sin2θcos2θ=1-sinθcosθ2=1-2sinθcosθ22=1-sin22θ4Now we know that sinθ lies between -1 to 1-1≤sinθ≤1So 0 ≤sin2θ≤1or 0 ≤sin2θ4≤14Now multiplying by -1 , then sign will get changed
-14≤-sin2θ4≤0Addding 11-14≤1-sin2θ4≤134≤1-sin2θ4≤1or 34≤A≤1Proved

If A = cos2 theta + sin4 theta, then prove that for all the values of theta, 3/4< equal to A < equal to 1.

If A = cos² theta + sin⁴ theta, then prove that for all the values of theta, 3/4< equal to A < equal to 1.