Home
Class 10
MATHS
If p sin^(3)alpha+qcos^(3)alpha=sinalpha...

If `p sin^(3)alpha+qcos^(3)alpha=sinalphacosalpha` and `p sinalpha - q cos alpha=0,` then prove that : `p^(2)+q^(2)=1`

Text Solution

Verified by Experts

`"p sin"alpha -qcosalpha=0`
`rArr p sin"alpha=qcosalpha` ...(1)
`:. "p sin" alpha+qcos^(3)alpha=sinalphacosalpha`
`rArr("p sina"alpha)sin^(2)alpha+qcos^(3)alpha=sinalphacosalpha`
`rArr(qcosalpha)sin^(2)alpha+qcos^(3)alpha=sinalphacosalpha` [from(1)]
`rArrqcosalpha(sin^(2)alpha+cos^(2)alpha)=sinalphacosalpha`
`rArr qcosalpha=sinalphacosalpha` (`:'sin^(2)alpha+cos^(2)alpha=1`)
`rArr q=sinalpha` ...(2)
Put `q=sinalpha` in equation (1)
`"p sin"alpha=sinalphacosalpha`
`rArrp=cosalpha` ...(3)
Now , `p^(2)+q^(2)=cosalpha+sinalpha=1`
Promotional Banner

Topper's Solved these Questions

  • INTRODUCTION TO TRIGONOMETRY

    NAGEEN PRAKASHAN|Exercise Problems From NCERT/exemplar|29 Videos

  • INTRODUCTION TO TRIGONOMETRY

    NAGEEN PRAKASHAN|Exercise Exercise 8a|20 Videos

  • CONSTRUCTIONS

    NAGEEN PRAKASHAN|Exercise PROBLEMS FROM NCERT/ EXEMPLAR|10 Videos

  • LINEAR EQUATIONS IN TWO VARIABLES

    NAGEEN PRAKASHAN|Exercise Revision Exercise Long Answer Questions|8 Videos

Similar Questions

Explore conceptually related problems

If p&q are lengths of perpendicular from the origin x sin alpha+y cos alpha=a sin alpha cos alpha and x cos alpha-y sin alpha=a cos2 alpha, then 4p^(2)+q^(2)

If p and p' are the distances of the origin from the lines x "sec" alpha + y " cosec" alpha = k " and " x "cos" alpha-y " sin" alpha = k "cos" 2alpha, " then prove that 4p^(2) + p'^(2) = k^(2).

If sinalpha+cos alpha=sqrt2 cos alpha then prove that tan 2alpha=1

If tan alpha+sin alpha=p and tan alpha-sin alpha=q tehn p^2-q^2=

If the line ax+by=1 passes through the point of intersection of y=x tan alpha+p sec alpha,y sin(30^(@)-alpha)-x cos(30^(@)-alpha)=p and is inclined at 30^(@) with y=tan alpha x, then prove that a^(2)+b^(2)=(3)/(4p^(2))

Evaluate : (cos^3alpha + sin^3alpha)/(1-sinalphacosalpha)

If sin.alpha/2+cos.alpha/2=1.4, then: sinalpha=

prove that (sin alpha+sin3 alpha+sin5 alpha)/(cos alpha+cos3 alpha+cos5 alpha)=tan3 alpha

If cos alpha + cos 2alpha = p , sin alpha + sin 2alpha = q then eliminate alpha .