Trigonometric Identities and Equations
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Direct answer
All trigonometry rests on sin^2 x + cos^2 x = 1 with its cousins 1 + tan^2 x = sec^2 x and 1 + cot^2 x = cosec^2 x, the double and triple angle expansions, and the sum-to-product conversions. Solving a trigonometric equation means quoting the general solution: sin x = sin alpha gives x = n pi + (-1)^n alpha, cos x = cos alpha gives x = 2n pi ± alpha, and tan x = tan alpha gives x = n pi + alpha, n an integer.
What you must remember
- Pythagorean identities: sin^2 x + cos^2 x = 1; 1 + tan^2 x = sec^2 x; 1 + cot^2 x = cosec^2 x — the base layer.
- Double angle: sin 2x = 2 sin x cos x; cos 2x = 1 - 2 sin^2 x = 2 cos^2 x - 1 = (1 - tan^2 x)/(1 + tan^2 x); tan 2x = 2 tan x/(1 - tan^2 x).
- Triple angle: sin 3x = 3 sin x - 4 sin^3 x; cos 3x = 4 cos^3 x - 3 cos x — the substitute when a cubic in sin or cos appears.
- Sum-to-product: sin C + sin D = 2 sin((C + D)/2) cos((C - D)/2); sin C - sin D = 2 cos((C + D)/2) sin((C - D)/2); cos C + cos D = 2 cos((C + D)/2) cos((C - D)/2); cos C - cos D = -2 sin((C + D)/2) sin((C - D)/2).
- Product-to-sum: 2 sin A cos B = sin(A + B) + sin(A - B), with companion forms for sin A sin B and cos A cos B.
- General solutions as above; a cos x + b sin x = c is solvable only when |c| <= sqrt(a^2 + b^2) — write the left side as R sin(x + phi) with R = sqrt(a^2 + b^2).
- Range: a sin x + b cos x assumes exactly the values in [-sqrt(a^2 + b^2), sqrt(a^2 + b^2)] — maximum and minimum without calculus.
Common confusion
Squaring fabricates roots: check every candidate in the original equation; only survivors count. Dividing by an expression that can vanish (cos x or sin x) silently deletes solutions — move everything to one side and factorise. And the general solution carries every integer n; quoting only the principal value answers a different question.
Exam-focused takeaway
JEE Main asks direct general solutions, identity-based simplifications and the range of a sin x + b cos x as quick numericals. JEE Advanced prefers equations reduced to products of factors, substitutions turning the equation into a quadratic in sin x or tan x, and identities fused with inverse functions or calculus. Compress with identities, factorise into zero-products, solve each factor, then verify.
Frequently asked questions
What is the general solution of sin x = sin alpha?
x = n pi + (-1)^n alpha; for sin x = 1/2, alpha = pi/6, giving pi/6, 5pi/6 and periodic extensions.
Why must solutions be checked after squaring?
Squaring mixes the signs of the two sides, so candidates satisfying the squared equation may violate the original; only verified ones count.
What is the range of 3 sin x + 4 cos x?
[-5, 5], since sqrt(3^2 + 4^2) = 5; the general rule is ±sqrt(a^2 + b^2).
Which formulas convert sums into products?
The sum-to-product set above; run them in reverse when a product needs integrating or simplifying.
What is cos 2x in terms of tan x?
(1 - tan^2 x)/(1 + tan^2 x), with companions 2 cos^2 x - 1 and 1 - 2 sin^2 x — pick the matching form.